# Introduction to Jitter, Noise, and Signal Integrity at High-Speed

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## 1.3 Signal and Statistical Perspectives on Jitter and Noise

We will first talk about the limitations and drawbacks of peak-to-peak-based metrics for jitter. Then we will discuss why the jitter component method of quantifying jitter is better and more accurate and should be used to describe and quantify statistical processes such as jitter and noise.

### 1.3.1 Peak-to-Peak and Root-Mean-Square (RMS) Description

For many years, jitter was quantified by peak-to-peak value and/or standard deviation (1 σ or rms) of the entire jitter histogram or distribution. It is now widely realized that this can be very misleading. In the presence of random and unbounded jitter or noise (such as thermal noise or shot noise), expected peak-to-peak value is a monotonically increasing function of statistical sample size. Peak-to-peak value is a useful parameter for bounded jitter or noise but not for unbounded ones. Similar problems occur with the standard deviation calculation. In the presence of bounded, non-Gaussian jitter or noise, the total jitter or noise histogram or distribution is not a Gaussian, and the statistical standard deviation or rms estimation does not equal the 1 σ of the true Gaussian distribution. Therefore, the latter is the correct quantity to describe a Gaussian process or distribution. Using standard deviation or rms based on the total jitter or noise histogram statistics "inflates" the true 1 σ value for Gaussian process.

To demonstrate the incorrect usage of statistical peak-to-peak in the presence of unbounded Gaussian jitter or noise, we start with a single Gaussian distribution via Monte Carlo method. We determine the peak-to-peak value for a given sample size N that is monotonically increasing and then plot the peak-to-peak value as a function of sample size. Figure 1.10 shows the results, clearly demonstrating the monotonicity trend. Figure 1.10 Peak-to-peak value plotted as a function of number of samples (N). The histogram distribution is a Gaussian, and the 1 σ of the Gaussian equals 0.03 UI (bit clock period).

To demonstrate how different a statistical standard deviation or rms and 1 σ of a Gaussian distribution can be, we assume that the histogram distribution has a bimodal distribution that is the superimposition of two identical Gaussians with different mean positions. Each peak corresponds to a single Gaussian mean position. Then standard deviation for such a bimodal distribution is 1.414 times (or 41.4% larger than) the true Gaussian 1 σ value when they are well separated (10 σ apart).

As the goal becomes to completely grasp the jitter or noise process, as well as to quantify the overall distribution and its associated components and root causes, the simple parameter-based approach to jitter or noise becomes insufficient and invalid. What is needed is the distribution function such as probability density function (PDF) and its associated component PDFs. Those PDFs not only give the overall description for jitter or noise statistical process, but also give the corresponding root causes.

### 1.3.2 Jitter or Noise PDF and Components Description

Jitter or noise is a complex statistical signal and therefore can have many components associated with it. We will focus on jitter, but the same concept applies well to noise. In general, jitter can be split into two components: deterministic jitter (DJ) and random jitter (RJ). The amplitude of DJ is bounded, and that of RJ is unbounded and Gaussian. This classification scheme is the first step in jitter separation.14

Jitter can be further separated after the first-layer splitting, as shown in Figure 1.11. Within deterministic jitter, jitter can be further classified into periodic jitter (PJ), data-dependent jitter (DDJ), and bounded uncorrelated jitter (BUJ). DDJ is the combination of DCD and ISI. BUJ can be caused by crosstalk. Within random jitter, jitter can be single-Gaussian (SG) or multiple-Gaussian (MG). Each jitter component has some specific corresponding root causes and characteristics. For example, the root cause of DJ can be a bandwidth-limited medium, reflection, crosstalk, EMI, ground bouncing, periodic modulations, or pattern dependency. The RJ source can be thermal noise, shot noise, flick noise, random modulation, or nonstationary interference.

A similar type of noise component tree classification can be developed, as shown in Figure 1.12.

Most of the component concepts for jitter and noise are symmetrical, except DCD, which does not have a noise counterpart. Also, the same type of jitter and noise component may or may not be correlated.