Section: Chapter 5. Jitter and Noise Separation and Analysis in the Statistical Domain
5.2. Jitter Separation Based on PDF
5.2.2. DJ PDF Determination Through Deconvolution
The Tailfit method introduced in section 5.2.1 determines the RJ PDF and DJ pk-pk value given the total jitter PDF. It also enables the extrapolation of jitter PDF from its higher-probability level to a lower-probability level. For example, if the total jitter PDF measured is at a probability level of 10–8, and the RJ PDF is determined, the total jitter PDF can be extrapolated to 10–12 or smaller for BER and TJ estimation required for most serial data communications.
However, what Tailfit does not give is the DJ PDF function directly, yet it is very important to understand and determine the nature and causing mechanism for DJ process. Because of this, the next section introduces a deconvolution-based method for determining the DJ PDF.
5.2.2.1. Deconvolution Theory
We derived the relationship between the total jitter PDF and its DJ and RJ PDFs via equation 4.1 of convolution by assuming that DJ and RJ are independent. This assumption can be justified, because DJ and RJ come from independent sources. In this section, we try to extract the DJ PDF from the TJ PDF, given that RJ PDF is known. A "blind" deconvolution is defined as a deconvolution process in which both DJ and RJ PDFs are unknown and are to be determined given the TJ PDF. Although from a pure theoretical standpoint a
"blind" deconvolution may still be possible, the accuracy and uniqueness of a "blind" deconvolution is generally poor, and that is not what we are interested in here.
We will start with the multiple Dirac-delta model of equation 3.9 and extend it as the general model for bounded DJ PDF. It is represented by the following:
Equation 5.5
Clearly, Δtn represents the location of the nth delta pulse of the DJ PDF, and P(Δtn) represents the probability in this jitter location. Δt1 and ΔtN represent the first (minimum) and last (maximum) time locations for the DJ PDF, respectively.
The TJ PDF is then given by the convolution between DJ PDF and RJ PDF, as defined by the following equation:
Equation 5.6
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Given TJ PDF fTJ(Δt), the first step is to obtain the RJ PDF via Tailfit, as discussed in section 5.2.1.
With a proper sampling scheme for both the TJ PDF and the RJ PDF, one of the approaches is to represent the convolution in a matrix form:
Equation 5.7
where T, R, and D are matrixes representing TJ, RJ, and DJ PDFs, respectively.
Clearly, if there exists an inverse matrix R–1, a true solution to equation 5.7 can be obtained:
Equation 5.8
More generally, a pseudoinverse for R can be obtained. Assuming that the inverse of (R' R) exists, R+ is given as
Equation 5.9
where R' is the matrix transposition of R. Then the least-squares solution to equation 5.8 can be written as
Equation 5.10
In other words, D+ satisfies the following:
Equation 5.11
For a detailed discussion of this matrix-based deconvolution, refer to [4].
5.2.2.2. Deconvolution Simulations
Numerical studies are performed to demonstrate the deconvolution algorithm introduced in the preceding section. The simulation is set up as follows: First, in the forwarding problem, a hypothesized DJ PDF is convolved with a known single Gaussian RJ PDF. Thus, the test-case TJ PDF is obtained. Next, the RJ matrix R is constructed by using the known RJ PDF, and then the DJ PDF is estimated by using the matrix method introduced. Finally, the recovered or deconvolved DJ PDF is compared with the original hypothesized DJ PDF.
In addition, the recovered TJ PDF is also compared with the assumed TJ PDF. We will study two distinct DJ PDFs, triangular and arbitrary shape, and present the simulation results accordingly.
5.2.2.2.1. Triangular DJ PDF
We will start with a triangular DJ PDF. To accommodate both theoretical and practical interests, two cases are considered: TJ PDF without statistical fluctuation, and TJ PDF with fluctuation. Figure 5.4 shows the results. In this figure, (a) is for "noiseless"
deconvolution, and (b) is for "noisy" deconvolution. In both cases, the estimated DJ PDF is a dotted line. Note that in (b), DJ PDF is not plotted, because it is the same as in (a). Instead, "noisy" TJ PDF (lower panel), original TJ, and "recovered" TJ PDFs (middle panel) are shown.
Figure 5.4. DJ PDF estimation when the original DJ PDF is a triangular function.
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We conduct the simulation in the context of serial data communication. The DJ PDF pk-pk value used is 0.2 UI, and the RJ Gaussian has σ = 0.33 UI. In the "noiseless" case, no TJ PDF smooth is needed. We see that the estimated DJ PDF reconciles with the original DJ PDF well. In the "noisy" case, TJ PDF needs to be smoothed before the deconvolution algorithm can be used. The general shape of the original DJ PDF is reserved in the estimated DJ PDF. Even though extra small "ripples" are introduced due to the statistical fluctuation in the TJ PDF, we have found that "recovered" TJ PDF obtained through convolution between estimated DJ and the assumed RJ PDF still agrees with the original "noiseless" TJ PDF.
5.2.2.2.2. Arbitrary DJ PDF
Figure 5.5 shows simulation results for an arbitrary DJ PDF with the same pk-pk as the triangular DJ PDF from the preceding section.
In this case, DJ PDF pk-pk and RJ PDF rms are kept unchanged. The agreement between the estimated and original DJ PDFs is even better for the arbitrary DJ PDF compared with that when the DJ PDF is triangular. This is because the arbitrary PDF we used does not have any "first-order" derivative discontinuity, although a triangular PDF does.
Figure 5.5. Similar to Figure 5.4, but for a rectangular DJ PDF.
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We have shown that for both triangular and arbitrary DJ PDFs, the matrix-based deconvolution method introduced in section 5.2.2 does a fairly good job of recovering the DJ PDF embedded in the TJ PDF, for both "ideal" and "practical" TJ PDFs. For a practical TJ PDF in which statistical fluctuation is present, a smooth process is needed to get a good DJ PDF recovery. The method can tolerate 10%
statistical fluctuation while still recovering a nearly perfect DJ PDF.
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Book: Jitter, Noise, and Signal Integrity at High-Speed
Section: Chapter 5. Jitter and Noise Separation and Analysis in the Statistical Domain
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Information Theory Computer Science Mike Peng Li Prentice Hall Jitter, Noise, and Signal Integrity at High-Speed