C APACIDAD PREDICTIVA DE LAS VARIABLES ANTROPOMÉTRICAS Y ANALÍTICAS PARA EL

PJ and RJ separation in the frequency domain will be discussed under the assumption that DDJ spikes have been removed from either the Fourier spectrum or PSD functions, as discussed in the preceding section. Figure 6.5 shows a PSD function without any DDJ spikes.

Figure 6.5. This PSD function contains PJ, RJ, and BUJ. DDJ spikes have been removed.

6.3.1. Based on the Fourier Spectrum

The Fourier spectrum in this case contains PJ, BUJ, and RJ. PJ appears as spikes or spectral lines. The PJ identification is based solely on its magnitude relative to the neighboring RJ and BUJ background. Any spectral lines that satisfy the following condition:

Equation 6.21

are identified as a PJ. The definitions of N and σ_FS are similar to those defined in equation 6.12. Many PJs may be identified. To get the overall PJ PDF from many individual PJs, PJ phase information is needed. PJs follow the superimposition rule, so the overall PDF can be established through the following relation:

Equation 6.22

where Δt_{PJ_l}, f_l, and φ_l are the peak value, frequency, and phase, respectively, for an individual lth PJ, and the summation is done over all the L PJs, with sampling times of t₁ to t_N.

When the PJs are identified and estimated, they can all be removed from the spectrum. However, Fourier spectrum is not PSD.

Approximated conversion of Fourier spectrum to PSD must be done through equation 6.4 and will not result in an accurate RJ estimation, as discussed in section 6.1.3.1. The RJ rms can be estimated via the following equation over an interested frequency range between f_L and f_H:

Equation 6.23

As soon as the RJ rms value is determined, its PDF is a Gaussian with the same rms or sigma value, assuming that the PSD is white.

Broadband BUJ is hard to separate from the RJ in the Fourier spectrum domain in general. As such, the RJ estimation may be inflated due to the presence of broadband BUJ in the Fourier spectrum.

One exception is that BUJ presence is under a controlled experiment or operation. For example, if the BUJ is caused by crosstalk from neighboring channels, BUJ can be estimated through two measurements. One uses the neighboring channel in quiescent mode and measures the PSD of RJ. Another method uses the neighboring channels in active mode and measures the PSD of RJ and BUJ. Because PSD follows the superimposition, the BUJ PSD can be estimated by taking the difference of the two PSDs measured. The result is the BUJ PSD estimation.

6.3.2. Based on PSD

There are similarities and differences between Fourier spectrum and PSD-based jitter separation. The PJ identification is similar to using the Fourier spectrum. Any spectral lines that satisfy the following condition:

Equation 6.24

are identified as a PJ. The definitions of N and σ_PSD are similar to those defined in equation 6.13. Many PJs might be identified.

However, no phase information is available in the PSD, because power is a scalar, not a vector. To get the overall PJ PDF from many individual PJs, the PJ phase information needs to be assumed, unlike the case of Fourier spectrum-based PJ separation. If there are many independent PJs, it may not be a bad idea to assume that their phases are randomly distributed, as shown in section 3.1.2. Under this assumption, the overall PJ PDF can be established through the following relation:

Equation 6.25

where Δt_{PJ_l}, f_l, and φ_l have the same meanings as in equation 6.20, except here the phases φ_l are assumed to be randomly and uniformly distributed, rather than determined, as in the case of Fourier spectrum-based separation.

The RJ PSD is obtained by removing all the PJs identified. The RJ rms can be estimated via the following equation over an interested frequency range between f_L and f_H:

Equation 6.26

This is the only accurate and correct way to estimate RJ rms over a certain frequency band. As soon as the RJ rms value is determined, its PDF is a Gaussian with the same rms or sigma value if the PSD is a Gaussian.

As we have mentioned, broadband BUJ is hard to separate from the RJ in the PSD domain too. Therefore, the RJ estimation can be inflated because of the presence of broadband BUJ in the PSD.

As mentioned in section 6.3.1, broadband BUJ can be determined with two PSD measurements and by estimating the difference residual if its presence can be controlled.

6.3.3. Based on Time-Domain Variance Function

PJ and RJ can be separated in the time domain given that jitter time record is available. We will start with the simplest model for PJ and RJ (DDJ has been separated at this stage), taking the sum of several sinusoidal signals (PJs) and the additive Gaussian white noise (RJ):

Equation 6.27

where Δt_{PJ_l}, f_l, and φ_i are the amplitude, frequency, and initial phase of the PJs, respectively. Δt(t) denotes the jitter signal, and Δt_RJ(t) is the zero mean Gaussian white noise with a variance of .

The goal here is to accurately estimate the number of PJs (L), their amplitudes and frequencies (Δt_{PJ_l}, f_l), and the RJ variance ( .).

The mean, variance, and autocorrelation function for Δt(t) can be written as follows:

Equation 6.28

where δ represents the Dirac delta function and τ is the time lag for the autocorrelation. In this case, the mean and variance are constant, and the autocorrelation is a function of only the time lag. In other words, all of them are independent of the time translation.

From the theoretical expression of the autocorrelation R_Δt (τ) in equation 6.28, jitter variance (RJ variance) can easily be estimated by averaging the integral of the autocorrelation function.

There are a few ways to solve equation 6.28 to find the solutions for PJ and RJ parameters. Two methods are worth mentioning because they are commonly used in mainstream digital signal processing. The first method converts the equation 6.28 into a matrix format and solves it with eigenvalue and eigen function. (Refer to ^[4] for more details.) The second method solves equation 6.28 through iterative optimization. We will discuss this method in depth here because it is relatively intuitive.

The variance record of the overall PJ and RJ relates to its autocorrelation function through

Equation 6.29

where is a constant representing the overall energy of the random process. is given by the second part of the equation 6.28 equation sets. If we rewrite the variance record of equation 6.29 in a discrete form as

Equation 6.30

the optimizer approach is fairly intuitive. Consider equation 6.30. We will "peel off" the sinusoids one by one. This is an uninvariant search process, assuming that the parameters are not interacting. In our case, let us define a cost function or gauge function:

Equation 6.31

By fixing arbitrary initial amplitude, we first sweep through a frequency range to find an f_e that minimizes E. Then we search through an amplitude range to find a Δt_{PJ_e} that minimizes E. With both amplitude and frequency known, we can remove one sinusoidal from the variance record. We can repeat this process until all the sinusoids are removed. The remainder then is pure noise.

Figure 6.6 shows an example of PJ and RJ separation based on time-domain variance function through the optimization method. This example has seven PJs and one RJ in the variance function. All seven PJs are separated in sequence, with both magnitude and frequency uniquely determined. The final residue in the variance function represents the RJ components. The determined PJs and RJ are very close to the expected values.

Figure 6.6. Time-domain PJ and RJ separation through the optimization method. The PJ+RJ variance function is shown at the top, and the separated RJ variance function is shown at the bottom.

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

Section: Chapter 6. Jitter and Noise Separation and Analysis in the Time and Frequency Domains

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