CAPÍTULO IV. RESULTADOS
1. E STUDIO DE PREVALENCIA
Chapter 5 introduced jitter separation in the statistical domain based on jitter PDF or CDF. Because jitter is fundamentally a statistical signal process, it can also be handled in the time or frequency domain. This chapter focuses on the time-frequency domain jitter separation.
6.1.1. Jitter as a Function of Time
Chapter 1, "Introduction," mentioned that jitter is any time deviation Δt referenced to an ideal timing signal, such as an ideal bit clock, used in digital communication. Δt may contain all the jitter components. Furthermore, jitter can be measured or observed only when an edge transition exists. This means that jitter is also a function of sampling times—namely, Δt(tn). Because Δt(tn) can be measured m times for a given time location tn, we will represent a generic jitter in the form of Δtm(tn). Using this concept, we can represent the instant jitter in terms of its components of data-dependent jitter (DDJ), periodic jitter (PJ), bounded uncorrelated jitter (BUJ), and random jitter (RJ) using the following equation:
Equation 6.1
Many mathematical operations can be applied to time-domain equation 6.1 to estimate jitter components of various kinds.
6.1.2. Jitter as a Function of Frequency
If a time-to-frequency domain operation or transfer is made to equation 6.1, the jitter frequency representation is obtained. The transformation operation can be a Fourier Transformation (FT), a Discrete Fourier Transformation (DFT), or a Laplace Transformation (LT), and so on. The frequency domain representation has a separate magnitude and phase part if a real frequency state variable is used.
If a complex frequency s is used as the state variable, as in the case of LT, one representation is enough, because both magnitude and phase are contained in the complex s domain. For simplicity, we will use an FT operation and show only the magnitude part of the frequency domain representation. In the actual implementation of the time-frequency domain transformation, a fast operation such as a Fast Fourier Transformation (FFT) may be used.
6.1.2.1. Direct FT Spectrum
Performing an FT operation on both sides of equation 6.1, we get the following frequency domain representation for jitter and its components:
Equation 6.2
[View full size image]
Each term in equation 6.2 represents the spectrum (first order) of its corresponding jitter. We will use the uppercase to represent the jitter spectrum in the frequency domain. Equation 6.2 can be rewritten as follows:
Equation 6.3
where l is the subindex of the discrete frequency. Even though the features and characteristics of overall jitter are complicated, its components have certain and unique characteristics. For example, periodic jitter shows up as spikes or spectral lines in the frequency domain, making it easier to identify and quantify this jitter component compared with a time-domain analysis. In another example, where DDJ is associated with a repeating pattern, DDJ also shows up as spikes or spectral lines. Their frequency is integer multiples of the pattern frequency fpatt, where fpatt = 1/(Npatt*UI) and Npatt is the length of the pattern in terms of UI. BUJ and RJ show up as broadband bounded background noise. Figure 6.1 shows the jitter spectrum containing all jitter components.
Figure 6.1. Magnitude of jitter Fourier spectrum, with all the components present. The DDJ frequency satisfies fDDJ = n*fpatt, where n is an integer and fpatt is the pattern length associated frequency.
Obviously, DDJ and PJ belong to the so-called "narrow-band" class. The major distinction between DDJ and PJ is that the DDJ frequency satisfies fDDJ = n*fpatt, whereas a PJ does not satisfy this relationship in general. However, when fDDJ = fPJ, DDJ and PJ cannot be separated in this jitter spectrum-based separation method. There are two types of BUJ: "narrow-band" and high-magnitude, and "broadband" and low-magnitude. Obviously, "narrow-band" BUJ is indistinguishable from regular PJ unless its root cause is known beforehand. "Broadband" BUJ is indistinguishable from RJ in general, unless its magnitude and frequency range are known beforehand, or the RJ spectrum shape and magnitude are known beforehand.
To estimate jitter energy within certain frequency bands, PSD is needed, and that is different from the Fourier spectrum. In practice, you may use the Fourier spectrum to estimate the PSD over a time period T by considering the following:
Equation 6.4
However, it has been shown[1] that equation 6.4 cannot approach the true PSD S(f), even when T approaches infinity. Thus, the Fourier spectrum does not provide a rational linkage to PSD for a given random process. Using equation 6.4 to estimate PSD and subsequent jitter energy is an approximation; it does not stand on solid theoretical ground or warrant necessary accuracy.
6.1.2.2. Jitter PSD
URL http://access.proquest.safaribooksonline.com/9780132429610/ch06lev1sec1
For a random process, what is interesting is how its energy is distributed over the frequency, not the spectrum that is phase-dependent.
Section 2.5.5 in Chapter 2, "Statistical Signal and Linear Theory for Jitter, Noise, and Signal Integrity," established the math foundation for estimating the PSD for a random process under the condition of wide-sense stationary (WSS). We need to start with the estimation of autocorrelation function to calculate the PSD. Taking the autocorrelation function operation of equation 6.1, we get the following:
Equation 6.5
[View full size image]
where RiJ_jJ are the cross-correlation functions between various jitter types. Because the sources of those different jitter types are distinctly different, it is not unreasonable to assume that they are independent or uncorrelated. The means for PJ, BUJ, and RJ are 0 by
definition. Therefore, the cross-correlation functions are 0s—namely, . Because of this property, equation 6.4 becomes
Equation 6.6
Then, applying the FT to both sides of the equation, we get the following:
Equation 6.7
Because an FT of the autocorrelation function gives rise to the PSD, equation 6.6 yields the following:
Equation 6.8
Equation 6.8 suggests that the overall jitter PSD equals the sum of individual component jitter PSDs. Superimposition works in this case because those jitter components are all independent and uncorrelated, and all the cross-spectral density vanished.
User name: CSU San Diego
Book: Jitter, Noise, and Signal Integrity at High-Speed
Section: Chapter 6. Jitter and Noise Separation and Analysis in the Time and Frequency Domains
No part of any chapter or book may be reproduced or transmitted in any form by any means without the prior written permission for reprints and excerpts from the publisher of the book or chapter. Redistribution or other use that violates the fair use privilege under U.S. copyright laws (see 17 USC107) or that otherwise
violates these Terms of Service is strictly prohibited. Violators will be prosecuted to the full extent of U.S. Federal and Massachusetts laws.
Information Theory Computer Science Mike Peng Li Prentice Hall Jitter, Noise, and Signal Integrity at High-Speed