Jitter Total PDF and the Relationship to Its Component PDFs

This section discusses the detailed mathematical relationship between total jitter PDF and the PDFs of its components.

4.2.1. Overall Jitter PDF

As you saw in Figure 1.11 in Chapter 1, "Introduction," jitter can be classified as deterministic jitter (DJ) or random jitter (RJ) at the first layer of separation. It is reasonable to assume that DJ and RJ are independent, because they are caused by independent and different sources and mechanisms. Recall the theorem introduced in Chapter 2, "Statistical Signal and Linear Theory for Jitter, Noise, and Signal Integrity." It says that the joint PDF sum of two independent variables is the convolution of their own PDFs (see section 2.1.2.6 of Chapter 2, equation 2.36). This gives us the following overall jitter PDF given PDFs of DJ and RJ:

Equation 4.1

Here the state variable is timing jitter Δt, and the PDF is for either the first zero-cross at 0 UI or the second zero-cross at 1 UI, as shown in Figure 4.1. In the second-layer separation, DJ can be separated into components of DDJ, PJ, and BUJ, represented by the following equation:

Equation 4.2

Similarly, RJ can be separated into random Gaussian jitter (RGJ) and random higher-order jitter (RHJ), as represented by the following equation:

Equation 4.3

If we represent the overall jitter PDF in terms of its second-layer components, we have this equation:

Equation 4.4

Equation 4.4 is the same as equation 3.45, where we gave the final results without the derivation. Equation 4.4 suggests that the overall jitter PDF equals the convolutions among all its jitter components' PDFs. It provides the math foundation to estimate the overall jitter PDF if all components' PDFs are known. Conversely, if the overall PDF and some of the component PDFs are known, some of the component PDFs can be estimated via the inverse operation of convolution, called deconvolution, which we denote as *^–1. If the total jitter PDF f_TJ is known, and any four of the five components' jitter PDF are also known, the fifth component PDF is uniquely

determined via deconvolution. Using Equation 4.1 as an example, if the total jitter and random jitter PDFs are known, the deterministic jitter PDF can be estimated as follows:

Equation 4.5

A similar equation can be derived for the random jitter PDF if both deterministic and random jitter PDFs are known:

Equation 4.6

4.2.2. Convolution for Jitter PDFs

It is helpful to demonstrate how to carry out the convolution for jitter PDFs to get the overall total jitter PDF. To conduct the convolution operation, jitter component PDFs of f_DJ and f_RJ need to be known for the first-layer separated jitter DJ and RJ.

This subsection assumes that RJ is a Gaussian or white, as defined by equation 3.33, and ignores the higher-order random jitter effects.

As discussed in Chapter 2, DJ PDF has no fixed form due to the variety of possibilities of causing mechanisms for the DJ

subcomponents. However, as discussed in Chapter 3, "Source, Mechanism, and Math Model for Jitter and Noise," a DCD PDF is best approximated by a dual-Dirac delta model given by equation 3.7. Furthermore, a single-frequency PJ PDF can also be approximated by a dual-Dirac delta model. Considering the fact that those practical DCD and PJ component PDFs are close to a dual-Dirac delta function, as well as for math simplification and illustration purposes, we will assume that the DJ PDF is a dual-Dirac delta function.

The convolution between a DJ dual-Dirac delta PDF and an RJ Gaussian PDF can be carried out analytically, using the property in which the convolution of an arbitrary function with a Dirac delta function equals the linearly shifted arbitrary function itself.^[1] This yields the following TJ PDF:

Equation 4.7

D_t is the DJ PDF peak-to-peak value, and σ_t is the RJ sigma value. Graphically, this convolution operation can be represented as shown in Figure 4.2.

Figure 4.2. TJ PDF determination from its DJ and RJ PDFs via a convolution operation.

Note that the TJ PDF is a bimodal distribution having even "twin peaks." The distance between those twin peaks is the same as the peak-to-peak value of the DJ PDF. In the tail region of the TJ PDF, its shape is the same as RJ Gaussian tails. Cleary, TJ PDF has some traceable characteristics of DJ and RJ PDF in this example. Chapter 5, "Jitter and Noise Separation and Analysis in Statistical Domain,"

has more information about the "inverse" problem of determining DJ PDF and RJ PDF by knowing TJ PDF.

4.2.3. Jitter PDF in the Context of an Eye Diagram

Figure 4.1 has two jitter PDFs: one for the 0 UI time location, and one for the 1 UI time location. Both correspond to the crossing point voltage. Due to how an eye diagram is constructed, the timing jitter PDF for 0 UI crossing and 1 UI crossing is the same. In this context, the overall jitter PDF is two identical PDFs, as defined by equation 4.7, separated by 1 UI time distance. The overall timing jitter PDF has this form:

Equation 4.8

[View full size image]

Graphically, this TJ PDF in the context of an eye diagram looks like Figure 4.3.

Figure 4.3. TJ PDF that is composed of the identical TJ PDFs at the first 0 crossing and the second 0 crossing.

[View full size image]

Clearly, the eye opening critically relies on the shape and characteristics of the TJ PDF and UI value.

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Book: Jitter, Noise, and Signal Integrity at High-Speed

Section: Chapter 4. Jitter, Noise, BER (JNB), and Interrelationships

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Information Theory Computer Science Mike Peng Li Prentice Hall Jitter, Noise, and Signal Integrity at High-Speed