# System-Level Statistical Eye Diagram for Signal Integrity
## 2 Statistical Eye Diagram
### Multi-Level Signaling  
A binary number system expresses data with ZEROs and ONEs. Thus, each bit has a LOW or HIGH logic value. Non-return zero (NRZ) signaling is used by binary systems in data transmission. However, the NRZ signaling is inefficient in terms of bandwidth because a single bit is only transmitted per signal pulse. In order to overcome this inefficiency, multi-level signaling has been widely used [33]. Multi-level signaling transmits the data with multiple bits per signal pulse, i.e., 01 as shown in Figure 8. The advantage of multi-level signaling is it decreases the bandwidth needed for the same data throughput. In other words, the data rate can be lowered if multiple bits are transmitted. The decreased data rate is determined by how many bits are transmitted per signal. The pulse amplitude modulation (PAM) with 4 levels (PAM-4) transmits two bits per signal. The relationship between the number of bits per signal \(N_{bits-sig}\) and the number of levels \(N_{lev}\) is as follows:  
\[N_{lev}=\log_{2}N_{bits-sig}. \tag{14}\]  
Figure 8: Single bit responses (SBRs) in multi-level signaling [34].  
Figure 7: Statistical eye diagrams depending on equalizers: (**a**) non-equalized channel, (**b**) DFE, (**c**) pre-emphasis, (**d**) de-emphasis, and (**e**) CTLE [32].  
Equation (14) also shows the required bandwidth for signaling is inversely proportional to the number of levels in the multi-level signaling. Therefore, for the same data rate, the required bandwidth can be decreased in the case of multi-level signaling.  
Based on the above SBRs, the corresponding statistical eye diagrams can be constructed by consecutive convolutions. The multiple-level signaling uses the scaled input pulse. Thus, the eye diagram is constructed based on the scaled channel responses [34]. Multi-level signaling has no effect on probability in the statistical approach. Because the SBR is only modified during the multi-level signaling, the statistical approach itself is used as it is. That is, the statistical approach has the modified input in this case. Another factor that has to be considered in multi-level signaling is the conversion loss. Conversion loss is caused by scaling the input pulse. Multi-level signaling has various levels to represent the data with a pulse. Thus, the amplitude is decreased by 1/N. That is, the pulse has a small amount of energy compared to NRZ signaling. The loss caused by the scaling in multi-level signaling is defined as the conversion loss. Therefore, in the case of multi-level signaling, the conversion loss always has to be compared with the insertion loss. In other words, when the channel is less lossy, then multi-level signaling may ironically degrade data transmission performance. This is another reason a statistical eye diagram is needed for multi-level signaling in high-speed systems [35].  
Figure 9 shows the statistical eye diagrams for multi-level signaling with different probabilities [36], which means each level has a different probability. The gray coding is used to assign the symbol to decrease the bit error rate in the case of an erroneous symbol. In other words, the symbol error rate does not need to be the same as the BER. Thus, the gray coding maps the symbol to have a short Hamming distance. As a result of the gray coding, each logic might have a different probability. The authors in [36] proposed a statistical eye diagram with logic probabilities to consider the effect of the encoding with multi-level signaling.