### Introduction

Escalating bit rates have reduced the time extent of the unit interval, or bit. The consequence is that jitter-creating signal integrity effects can easily result in bit errors. Subsequently, quantifying jitter has become a requirement in almost every high-speed serial data communications protocol. Engineers must measure the amount of jitter due to transmitter and channel effects, and understand the contributions to the overall jitter of individual components. Modern digital storage oscilloscopes, especially at the high-end, include serial data analysis options that measure the jitter of acquired signals and extrapolate it using the *dual-Dirac model*. The goal is to predict the impact of very low probability jitter events that yield bit errors at small levels, such as 1 in 10^{12}.

The jitter analysis performed by these instruments can return a value for deterministic jitter (Dj) that is less than one if its components, including data dependent jitter (DDj) and periodic jitter (Pj). Figure 1 shows an example. In order to explain why this situation can occur, this application note reviews the concepts of deterministic jitter, data-dependent jitter and periodic jitter, briefly describes how they are measured, and then discusses two scenarios where Dj < DDJ or Pj. For a much more detailed description of the jitter calculation methodology, see reference [2].

### Jitter Hierarchy

Figure 2 shows the industry-accepted jitter hierarchy. Total jitter breaks down into random and deterministic components, which have the attributes of being either unbounded or bounded, respectively. Deterministic jitter (Dj) breaks down further into two buckets: Correlated to the data pattern, and uncorrelated to the pattern. Jitter that is bounded and correlated to the
data is "Data dependent jitter" (DDJ). Everything bounded and uncorrelated goes
into the bucket called "BUJ", including periodic jitter (Pj). From Figure 2, it is easy to see that data-dependent jitter and periodic jitter are a subset of deterministic jitter. Yet it is still feasible for Dj < DDj or Pj. How can this be possible?

### What is Deterministic Jitter?

Deterministic jitter is defined as jitter that is bounded, with a well-defined minimum and maximum extent. This is in contrast to random jitter, which is Gaussian in nature, and is unbounded. There are a variety of deterministic data sources, including data-dependent jitter (DDj) and periodic jitter (Pj). Figure 3 shows the jitter distribution due to a purely periodic jitter aggressor that is a single sinusoid. Periodic jitter is caused by clocks or other periodic sources that can modulate the transmitted edges, and, in the case of sinusoidal jitter with one frequency, yields a bowl-shape jitter distribution. (However, you would not see such an idealized histogram in a real signal; the inevitable presence of random jitter changes the distribution, and discussed later on.) Note that the histogram in the figure has a well-defined range: it is bounded. As such, this type of jitter is deterministic.

One further comment about deterministic jitter. There can be a large number of individual Dj sources, each contributing tiny amounts of bounded jitter. By virtue of the central limit theorem, these can add up to be random, Gaussian jitter, and are characterized as such.

### How is Deterministic Jitter (Dj) Calculated?

In high-end serial data analysis, Dj is calculated by performing a fit to the dual-Dirac jitter model. In this context, Dj is written much more precisely as Dj(delta-delta), or Dj(δδ). In fact, the nomenclature used to describe deterministic jitter is a source of confusion. When only written as "Dj", as is often the case, the fact that Dj is not "Dj(peak-peak)" can easily be forgotten and results misunderstood.

The dual-Dirac model describes jitter as two Dirac delta functions convolved with a Gaussian. Figure 4 shows a depiction of the dual-Dirac model formulation. The delta functions above μL and μR model deterministic jitter. Their separation in time (μL - μR) is Dj(δδ). The Gaussian distribution (dashed curve) models random jitter. Dj(δδ) is determined by fitting the cumulative distribution function (CDF) of jitter to the formula Tj(BER) = alpha(BER) * Rj + Dj(δδ). Alpha is a multiplier based on the user’s selection of BER. For example, for BER = 10-12, alpha = 14.07. For more info on the dual-Dirac model see the references.

Conceptually, the model's job is to make a prediction about jitter in the presence of a big data set, e.g. >1012 unit intervals. To do this job, the model extrapolates the tails of the random jitter to model its growth. (The breaks in the solid curve depicts how random jitter is extrapolated.) Since > 1012 unit intervals is much larger than can be acquired in a short period of time on any oscilloscope, many serial data standards committees have standardized on the dual-Dirac model for determining Tj(BER), Rj and Dj(δδ).

**The key point here is that Dj(δδ) is not calculated by summing individual components of deterministic jitter. Instead, it is a fit to a model. Consequentially, Dj(δδ) is not a "peak-to-peak" or "full scale" measure.**

Many engineers find the fact that Dj(δδ) is not a peak-peak measurement unsettling, and want to have a peak-peak value for Dj. Finding Dj(peak-peak) without an a priori knowledge of all deterministic jitter aggressors would be quite challenging. For example, looking at the jitter histograms in Figure 5, how would you go about removing the random component of jitter in order to get a peak-peak value for Dj? It might seem straightforward for the histogram on the left side, but it can get quite challenging for channels with "complex" jitter distributions like on the right.

### What is Data-Dependent Jitter?

The electrical characteristics of the channel can result in jitter that is observed to be dependent on the data pattern being transmitted. For example, the jitter on the last edge of the bit sequence 00001 can be different than after a different sequence, such as 11101. Reflections and frequency-dependent losses in the channel or interconnect cause this behavior, which is called inter-symbol interference (ISI). Figure 6 shows an example of ISI. The red trace is the signal. The blue trace is the "jitter track" waveform. The jitter track shows the measured jitter in time with the acquired signal. Note that the jitter track shows a repetitive pattern just like the signal. Each bit in the pattern has a jitter value that depends on the bit history. ISI is a type of deterministic jitter, and is bounded like the example of periodic jitter shown in Figure 3. However, this jitter is correlated to the data pattern, and is therefore "data-dependent jitter". Lastly, the other type of data-dependent jitter is due to variations in the crossing level used to identify the edge timing, an offset in the signal amplitude, or asymmetry in rise/fall times. This type of impairment is duty-cycle distortion (DCD).

### How is Data Dependent Jitter (DDj) Calculated?

Unlike the calculation of Dj(δδ), which is performed with both extrapolation and fitting steps, DDj is calculated directly from the acquired data. For a signal that is a repeating pattern, the time interval error measurements are analyzed for evidence of the repeating pattern. The “DDj Histogram” shown in Figure 7 provides the resulting jitter distribution. The full-scale range of this distribution is the DDj measurement. DDj breaks down into ISI and DCD; measurements for ISI and DCD derive from an analysis of the DDj histograms using positive and/or negative edges. See reference [2] for more information. DDj is "full-scale" or "peak-to-peak" measurement. Since DDj is bounded, using a pk-pk measurement for DDj is the correct approach. As such, the resulting DDj measurement does not include any unbounded jitter.

### What is Periodic Jitter?

As mentioned previously, periodic jitter is caused by clocks or other periodic sources that can modulate the transmitted edges. The subsequent jitter track oscillates, as can be seen in Figure 8. When the Pj from a single sinusoidal contributor dominates over all other jitter sources, a bowl-shaped distribution can be observed. In the presence other jitter sources such as a random jitter, ISI or multiple Pj contributors, it might be impossible to see a bowl-shaped jitter histogram. However, the jitter track might show clear evidence of a periodic jitter aggressor. Figure 8 shows the jitter track and histogram for two signals that are suffering from periodic jitter, with some random jitter mixed in.

### How is Periodic Jitter Calculated?

To isolate the periodic jitter from other sources, the data dependent jitter is first “stripped” using pattern averaging in order to yield a jitter track that contains only random and bounded-uncorrelated jitter (including Pj). A spectral analysis of this track is performed that finds and isolates Pj peaks from the background noise in the jitter spectrum. The peak-peak extent of the Inverse FFT of the Pj contributors becomes the Pj result. Figure 9 shows an example analysis. For more information, see reference [2].

### Dj(δδ) can be less than DDj or Pj

Since Dj(δδ) is a fit to a model and not simply a full-scale measurement like DDj or Pj, it should be apparent that comparing Dj(δδ) to DDj and Pj must be done with some caution. If Dj was a "full-scale" measurement (i.e. "Dj(pk-pk)"), then DDj and Pj would always be less than Dj. This would be true "by definition", since data-dependent jitter and periodic jitter are a subset of deterministic jitter in the jitter hierarchy. However, in the dual-Dirac model, Dj is not defined as Dj(pk-pk); instead, Dj(δδ) is a model-dependent parameter, with the consequence that Dj(δδ) can be less than DDj or Pj.

In general, models only match reality up to a point. In real-world signals, the actual jitter distribution almost **never** resembles the jitter distribution for the dual-Dirac model. This fact alone is enough to conclude that Dj(δδ) will not equal Dj(peak-peak). But why, in general, is Dj(δδ) < Dj(peak-peak)? And why, potentially, is Dj(δδ) < DDj and/or Dj(δδ) < Pj?

When the deterministic jitter distribution includes significant population internal to the extremes, as in Figure 10, then the convolution of the random jitter Gaussian and the deterministic jitter distribution yields a PDF (and subsequently, a CDF) that is a poor fit to the dual-Dirac model. The consequence is that when the fit to Tj = alpha * Rj + Dj(δδ) fit performed, the Dj(δδ) evaluates to a value that is "pulled in" from the actual Dj(peak-peak) value. Figure 10 shows this effect. The top distribution shows a high ratio of Pj to Rj. As the amount of random jitter increases, the convolution of the distributions (Gaussian for random jitter and sine PDF for periodic jitter) brings the peaks inward. Consequentially, when DDj (or Pj) is the primary source of deterministic jitter, then Dj(δδ) < DDj (or < Pj), and Dj(δδ) < Dj(peak-peak). The fact that the dual Dirac fit is performed out on the tails of the distribution also causes there to be some sensitivity to the amount of data acquired. It is important to acquire sufficiently long waveforms and repetitions of the pattern in order to avoid insufficient statistics.

### Example #1: Dj< Pj

Figure 11 shows such a result for the scenario where Dj(δδ) < Pj. The jitter table shows the results: Dj = 1.27 ps, Pj = 1.47 ps. This is a simulated waveform that could potentially be the output of a very low-jitter clock, but that is suffering from coupling to an external periodic aggressor.

Figure 12 shows how the distribution due to purely periodic jitter (green) compares to the actual PDF (red). Since the dual-Dirac model is based on two Dirac delta functions, one can expect the fit to place the delta functions somewhere near the two maxima of the red distribution. The white cursors shows the interval corresponding to the Dj(δδ). Note that it is well within the peak-peak extent of the green distribution, which illustrates the distribution due to the Pj contributor alone. The result is that Dj(δδ) < Pj. (Note: the vertical scales of the two histograms are not representative of the relative populations.)

### Example #2: Dj< DDj

Figure 13 shows the scenario where Dj(δδ) < DDj. In this case. The jitter table shows the results: Dj(δδ) = 34.97 ps, Pj = 36.706 ps. The overall jitter distribution shows the effects of ISI convolved with random jitter. Clearly, this distribution is a terrible fit to the dual-Dirac model, and the jitter distribution shows what appears to be many individual peaks.

Pattern analysis is used to understand the extent of the DDj. Figure 7 (page 4) shows the DDj histogram overlaid on top of the overall jitter distribution. For the same reasons as discussed for Example 1, Dj(δδ) evaluates to less than DDj; the positions of the 2 Dirac functions are pulled inwards due to the distribution’s poor fit to the dual-Dirac model.

### Validity of the Dual Dirac Model

One might ask the question, "Why use the dual-Dirac model if it isn't a good fit to real-world jitter distributions?" There are a few ways to answer this question. One "easy" answer is from the perspective of the instrument manufacturers: "Because our customers require that the oscilloscope use this model." Another answer, not entirely satisfying, is "There isn't a better industry-accepted one to use".

Even with its limitations, the model is still quite useful, and meets the needs of standards bodies. The model is well-defined and standardizes a method to understand total jitter as it relates to the probability of an occurrence of a bit error. It avoids using a peak-to-peak result for determining total jitter, which is a wise choice, since peak-to-peak results depend heavily on the amount of data acquired, are random due to the unbounded nature of random jitter, and do not predict the amount of jitter at a specific BER level. The dual-Dirac model is also usable by different instrument types, such as real-time oscilloscopes, sampling oscilloscopes and bit error rate testers.

### Conclusion

Deterministic jitter is a measurement that is easily misunderstood. Engineers often want it to describe peak-peak deterministic jitter, but determining Dj(peak-peak) is non-trivial, especially in the presence of complex jitter distributions caused by ISI and Pj. Instead, deterministic jitter is quantified via the dual-Dirac model, with the model dependent result Dj(δδ). Since the convolution of the Gaussian representing random jitter with the deterministic jitter can pull in the locations of the Dirac delta functions, Dj(δδ) < Dj(peak-peak). When Deterministic jitter is dominated by either data-dependent jitter (DDj, typically due to ISI) or periodic jitter (Pj), then the inequalities Dj(δδ) < DDj, Dj(δδ) < ISI and Dj(δδ) < Pj can occur.

### References

[1] "Fibre Channel - Methodologies for Jitter and Signal Quality Specification- MJSQ". T11, 5 June, 2005. http://www.t11.org (t11.org membership required.)

[2] "Understanding SDAIII Calculation Methods". Teledyne LeCroy Website, 2012, http://cdn.teledynelecroy.com/files/whitepapers/understanding_sdaiii_jitter_calculation_methods.pdf

[3] Stephens, Ransom, "Jitter Analysis: The dual-Dirac Model, RJ/DJ, and Q-Scale". Agilent Technologies Website, 2004. http://cp.literature.agilent.com/litweb/pdf/5989-3206EN.pdf