# System-Level Statistical Eye Diagram for Signal Integrity
## 2 Statistical Eye Diagram  
An eye diagram has the critical drawback of being time-consuming due to the principle of acquisition. An eye diagram is an overlapped waveform at a receiver. Thus, the reliability of an eye diagram is determined by how many bits are received. That is, more received bits are always required to avoid a biased result. If the target bit-error rate (BER) is \(10^{-N}\), then the required number of bits would be \(10^{N}\). The measured eye diagram might be achievable depending on the memory of the measurement instrument used such as an oscilloscope. However, a simulated eye diagram is different in terms of calculation and memory. In a simulation, a data bit has multiple points, i.e., 50, 64, or 100 to represent the continuous waveform within a unit interval (UI). Thus, the required number of bits is multiplied by the number of points in the UI. Therefore, a simulation requires a larger memory than that needed for eye diagram measurements. Another difference between simulated and measured eye diagrams is how they collect waveforms. The measured waveforms are recorded by a sampling and an analog-to-digital (A-to-D) conversion [10]. The recorded data are accumulated to construct an eye diagram. However, the simulated waveforms are calculated based on the channel response and input waveforms. The calculation includes consecutive additions in a transient simulation to calculate the inter-symbol-interference (ISI) noises [11]. The ISI noise is defined as the sum of each received bit which is a result of the degradation caused by the RC delay [12]. Each  
Figure 1: The eye diagram is a critical SI metric to show electrical degradation such as crosstalk between channels and insertion loss by parasitic resistance and capacitance. The eye diagram is obtained by overlapping the received waveforms; thus, it requires a significant amount of acquisition time. bit may have a value of either ZERO or ONE; thus, the ISI noise might be determined by a huge number of bits within the ISI noise. This leads to the introduction of the eye diagram estimation methods into an SI analysis. As can be seen from Figure 2, a peak distortion analysis (PDA) calculates the inner-most contour of an eye diagram, which is defined as the worst contour of an eye diagram [13]. The worst contour provides the EH value in the amplitude and the EW value in the sampling time. Thus, the PDA method has been widely used for an efficient SI analysis. However, the worst contour only provides simplified results, which leads to limited usage. In order to provide more information on the SI evaluation, the statistical eye diagram was introduced. The statistical approach provides probability distributions depending on either a sampling time or a sampling voltage. Hence, some postprocessing algorithms are applied to obtain further information such as the BER values from a statistical eye diagram. The BER values represent how many errors occur during data transmission. A system-level SI analysis typically includes a BER-based analysis [14]. The statistical approach requires a longer calculation time and more memories during the calculations. When it comes to the calculation, a PDA includes selective additions. However, the statistical approach includes consecutive convolutions to calculate the PDF. The above properties are critical drawbacks of the current eye diagram estimation methods. Nevertheless, the PDA and statistical approach are still faster and simpler than a transient simulation. The statistical approach includes the following steps to calculate the probability distributions depending on the sampling time [15]:  
1. Single-bit response (SBR) calculation;
2. Cursor separation from the SBR;
3. ISI calculation with the pre-and post-cursors;
4. Bit PDF calculation with the main cursors;
5. The above procedures are iterated with sampling times. The SBR is the channel response for a single bit, which is the attenuated and widened pulse caused by parasitic resistance and the capacitance of a high-speed channel.