DDJ Separation

This section discusses DDJ separation methods based on jitter time-domain record, frequency-domain spectrum, and PSD. In the end, we will discuss how to separate DDJ into its components of DCD and ISI.

6.2.1. Based on Jitter Time Function

By applying the average operation to equation 6.1 over the number of measurements M for all the sampling times indexed by n, we get the following:

Equation 6.9

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The average of all PJ, BUJ, and RJ gives rise to 0 if the sample size and record are large enough, because they are uncorrelated to the data pattern. DDJ is static, and the average will not change its value. Therefore, equation 6.9 becomes

Equation 6.10

Equation 6.10 gives a quantitative way to estimate DDJ from the generic jitter Δt time record in the time domain. In the context of data communication, the jitter test or estimation is generally performed using a certain repeating test pattern with a fixed number of transitions.

To put equations 6.9 and 6.10 into perspective, Figure 6.2 shows a digital waveform (a 20-bit long K28.5 pattern commonly used in speed testing) that has jitter components of DDJ, PJ, and RJ. DDJ is due to the limited bandwidth of the medium where the high-speed signal passes through. PJ and RJ are due to the amplitude modulation of both periodic and random noises to the signal.

Figure 6.2. A digital waveform with DDJ, PJ, and RJ at its 50% reference level.

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Performing the averaging for the waveform shown in Figure 6.2, we obtain the waveform shown in Figure 6.3, where pattern uncorrelated jitter PJ and RJ both average out. Therefore, the edge transition deviation relative to the ideal bit clock gives rise to DDJ estimation.

Figure 6.3. The averaged waveform of Figure 6.2. Note that PJ and RJ are averaged out so that edge transition deviation relative to the ideal clock is DDJ only.

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The DDJ PDF can be built based on the DDJ time record given by equation 6.10 through the histogram binning:

Equation 6.11

where Hist represents the histogram binning function.

Time-domain DDJ was first developed by ^[2].

6.2.2. Based on Fourier Spectrum or PSD

DDJ can be estimated in the frequency domain via Fourier spectrum or PSD. The criteria are that DDJ magnitude in the frequency

domain needs to be a few sigmas above the background RJ and BUJ spectrum, and DDJ frequency needs to be integer multiples of the pattern repeating frequency. In other words, when the following conditions are satisfied, a DDJ is identified. For Fourier spectrum-based identification, the criteria are as follows:

Equation 6.12

where σ_FS is the background rms for the Fourier spectrum and N is the threshold level. Typically N satisfies that N 3. When N = 3, a 99.97% statistical confidence level is warranted (see section 3.2.1.1). Fourier spectrum-based DDJ separation can also be found in ^[3]. For a PSD-based identification, the criteria are as follows:

Equation 6.13

where σ_PSD is the rms for the PSD, and the N threshold level is similar to the case of Fourier spectrum-based DDJ separation.

The number of DDJ spikes is determined by the pattern length, transition density, run length, and so on. If Δt_DDJ(f_l) is the peak value of DDJ at frequency f_l, the DDJ PDF will be

Equation 6.14

This is very similar to equation 6.11, in which DDJ is determined through time-domain average.

6.2.3. DCD and ISI Separation from DDJ

Before we get into DCD and ISI separation from DDJ, let us first review their definitions.

DCD is duty cycle distortion. DCD can be caused by variations in the reference voltage level used to determine the pulse width. It also can be caused by different propagation delays for positive and negative data transitions.

ISI is intersymbol interference. ISI is caused by a data path propagation delay that is a function of past data history and occurs in all finite bandwidth data paths.

DDJ is composed of both DCD and ISI.

Let us assume that the PDFs for DDJ rising and falling edges are f_{DDJ_r} and f_{DDJ_f}, respectively. We have the following relationship:

Equation 6.15

We define the maximum PDF corresponding jitter location as Δt_{max_r} and Δt_{max_f} for rising and falling edges, respectively. The ISI PDF is the DDJ PDF when Δt_{max_r} and Δt_{max_f} are identical because, by definition, if there is no DCD, rising- and falling-edge PDFs should be lined up. This can be achieved by the following math operation:

Equation 6.16

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when Δt_{max_f} Δt_{max_r}. and

Equation 6.17 [View full size image]

when Δt_{max_r} Δt_{max_l}.

Because DDJ is composed of DCD and ISI, and their PDFs are linked through the convolution process, DCD PDF can be obtained through the following deconvolution process:

Equation 6.18

where *^–1 denotes the deconvolution operation.

In the case when both rising- and falling-edge DDJ PDFs are symmetrical, with only one peak, DCD PDF is simply a dual-Dirac delta function represented by the following:

Equation 6.19

and its pk-pk value in this case is Δt_{max_f} – Δt_{max_r} for Δt_{max_f} Δt_{max_r}.

In the case when Δt_{max_f} Δt_{max_r}, we have

Equation 6.20

with a pk-pk value of Δt_{max_r} – Δt_{max_f}.

Figure 6.4 shows the relationship between DDJ, DCD, and ISI in a single graph. For simplicity, the rising-edge PDF and falling-edge PDF are identical and have only single peaks. We consider the case of Δtmax_f Δtmax_r. According to equation 6.16, ISI PDF is the superimposition of the rising- and falling-edge PDFs when their peaks are aligned. Because rising- and falling-edge PDFs are identical and have a pk-pk value of a, the ISI PDF derived from equation 6.16 also has the same PDF with the same pk-pk value a. With DDJ and ISI PDFs, DCD PDF can be estimated via equation 6.18 of deconvolution, yielding a dual-Dirac PDF for DCD.

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Figure 6.4. PDFs for a DDJ and its components of ISI and DCD. DDJ rising- and falling-edge PDFs are single-peak and symmetrical.

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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