# System-Level Statistical Eye Diagram for Signal Integrity
## 2 Statistical Eye Diagram
### Equalizer  
Equalizers compensate for the electrical degradation in either the frequency or time domains [23; 24; 25; 26]. The frequency-domain equalizer includes a continuous time linear equalizer (CTLE). The time-domain equalizers include the pre-/de-emphasis and decision feedback equalizer (DFE) as shown in Figure 6. The CTLE either amplifies the high-frequency energies or attenuates the low-frequency energies to make the channel response as flat as possible [27]. Typically, the CTLE is placed just before the receiver to directly compensate for the channel loss. Likewise, a pre-emphasis boosts the high-frequency energies and a de-emphasis attenuates the low-frequency energies in the time domain. Unlike the CTLE, the pre-/de-emphasis are placed at the transmitter. In other words, the CTLE mitigates the channel loss after the channel and the pre-/de-emphasis compensates for the channel loss before the channel. The DFE is a time-domain equalizer mitigating the loss after the channel [28; 29; 30; 31]. The DFE makes a bit decision based on the received waveform: LOW or HIGH in a binary system. And the equalizer mitigates the expected ISI noise based on the previous bits. As discussed earlier, the ISI noise caused by parasitic resistance and capacitance makes the channel response attenuated and widened. Thus, if the received bit is equal to ONE, the next bit would experience the ISI noise just created by the previous bit. The DFE attenuates the currently received waveform by its coefficient to mitigate the expected ISI noise. Figure 5: The statistical eye diagrams have different probability distributions depending on the 8B/10B and TMDS encoders [21]: (**a**) eye diagram without the encoding, (**b**) eye diagram with 8B/10B encoding, and (**c**) the eye diagram with TMDS encoding. As a result of the encoding, the probabilities of the bit transitions are different for both cases. In the case of 8B/10B encoding, the non-transition area has a lower probability. The change in the probability can be identified from the darker area in the statistical eye diagram and the same is true for, the case of TMDS encoding. Because the DFE operates in a digital manner, the effect of the DFE on an eye diagram has to be interpreted in a probability domain for a statistical eye diagram. The statistical eye diagrams are based on the equalized SBR. The conventional statistical approach is applied after the equalizers are applied to the SBR. That is, the equalized SBR replaces the SBR in the statistical approach. The probability distribution in the statistical eye diagram is determined by the SBR and the consecutive convolutions. Therefore, the equalizer only modifies the input of the statistical approach, not the algorithm. Figure 6 shows the equalized SBR depending on the equalizers. By definition, the equalized SBR by the pre-emphasis has a boosted channel response at the rising area [32]. Thus, the resultant waveform has boosted high-frequency energies compared to the non-equalized ones. Likewise, the de-emphasis has a dip at the falling area to mitigate the expected ISI noise. The CTLE also amplifies the high-frequency energies. The difference between the pre-emphasis and the CTLE is the amplifying area in the waveform. The pre-emphasis only locally amplifies the high-frequency signal at the rising area. In contrast, the CTLE amplifies the high-frequency signal in the overall area of the waveform. Figure 7 shows the statistical eye diagrams depending on the equalizers [32]. Because the color of the statistical eye diagram represents the probability, how the equalizer works on the eye diagram is identified. The equalized statistical eye diagram has a higher probability at the center of ONE and ZERO. Because the DFE equalizes the waveform based on the recovered data, the timing of the feedback signal determines which area has a higher probability. The pre- and de-emphasis make the probability higher after and before the center in the eye diagram, respectively. The pre-emphasis causes an additional peak in the waveform.