# System-Level Statistical Eye Diagram for Signal Integrity
## 4 Discussion  
The introduced previous works proposed statistical approaches to system-level eye diagrams. Studies on system-level statistical SI analyses still have to be addressed in the future to achieve the desired system performance evaluation in statistical and practical manners. The objective of a statistical SI analysis is to accurately evaluate a system in an efficient manner. The transient circuit simulation definitely has a limitation in terms of calculation and memory. Thus, a system-level analysis based on a circuit simulation is being replaced by a statistical SI analysis. However, a statistical SI analysis has a discrepancy due to the limited information of each system. The lack of information causes an impractical SI analysis. Therefore, the components in high-speed systems have to be interpreted to be integrated into a statistical eye diagram. In order to reflect the effect of the components in a statistical eye diagram, each component has to be modeled in terms of probability. Although this article reviews several previous studies, the technical trend drives the need  
Figure 11: Statistical eye diagram depending on the ECC. (**a**) The BCH code encodes the data bits in a bit-wise fashion, thus the effect of the BCH code on the eye diagram is not significant. (**b**) In contrast, the RS code encodes the data bits in a symbol-wise fashion, thus the RS code make the bit probability of ZERO higher [15].  
to equip systems with more techniques to improve SI and BER performances. Thus, as the systems evolve, further study on statistical approaches to high-speed systems have to be conducted.  
The introduced approach proposed here describes how each component of a high-speed system can be applied in a statistical approach. Typically, no high-speed system uses a single component to improve SI performance. Namely, all of the systems include several techniques. In other words, the above methods have to be considered for practical purposes. The combination of the techniques used is determined by the system designer. For the systems consisting of 8B/10B encoding and equalizers, the equalized SBR has to be applied to the statistical approach for 8B/10B encoding. If the system includes scrambling and an ECC, then the ECC has to be considered first. Because the purpose of the scrambling is to suppress the radiated EMI, the effects caused by the ECC have to be reflected first. Therefore, the priority of the high-speed component in the statistical approach can be defined like the above.  
A statistical SI analysis can be utilized in a BER analysis if post-processing is applied. A BER analysis typically is fulfilled with a shmoo plot [47]. The shmoo plot shows the BER values depending on the system conditions such as supply voltage, threshold voltage, and operating frequency. The shmoo plot has been especially used in the silicon industry. Because of the statistical characteristic of the shmoo plot, any of the probability density functions (PDFs) are applicable. The typical parameters such as process, voltage, and temperature (PVT) variation can be analyzed in a statistical SI analysis. In other words, a statistical SI analysis is expandable depending on the post-processing. Thus, a statistical SI analysis is valuable in terms of expandability. The above benefits show the importance of a statistical approach to an SI analysis.