# System-Level Statistical Eye Diagram for Signal Integrity
## 2 Statistical Eye Diagram
### Error-Correction Codes (ECCs)  
Error-correction codes (ECCs) modify the bit sequence to correct the errors in the received bits in a mathematical manner [40; 41]. The ECC encodes the data bits and generates a word [42; 43; 44]. The words, instead of the bits, are transmitted over a channel. The receiver corrects the erroneous bits during the decoding. The Bose-Chaudhuri-Hocquenghem (BCH) code corrects a single bit error in a word [45] and the Reed-Solomon (RS) code corrects multiple bit errors [46]. Their difference is due to the Galois Field which defines the  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{Data (before scrambling)} \\ \hline Bit index & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ Probability & 0.562 & 0.629 & 0.458 & 0.649 & 0.495 & 0.487 & 0.476 \\ \hline \hline \multicolumn{8}{c}{LFSR \(2^{7}-\)1 for scrambling} \\ \hline Bit index & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ Probability & & & & 0.5039 & & & \\ \hline \hline \multicolumn{8}{c}{Scrambled data (after scrambling)} \\ \hline Bit index & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ Probability & 0.500 & 0.499 & 0.500 & 0.499 & 0.500 & 0.500 & 0.500 \\ \hline \hline \end{tabular}
\end{table}
Table 1: ONE’s probabilities in the transient simulation [39].  
Figure 10: Statistical eye diagram depending on the scrambling. (**a**) When the biased data are given, it has a higher probability for either ONE or ZERO. The biased probability distribution is identified from the asymmetry of the probability. (**b**) After the scrambling, the corresponding eye diagram become symmetric which means the ONE and ZERO have the same probability [39].  
generator polynomial. To correctly estimate the statistical eye diagram for the ECC-encoded channel, generator polynomial-based statistical eye diagrams were introduced [15].  
As discussed, the ECC is based on the generator polynomial for the Galois field. Thus, the codeword transmitted over the channel is determined by the generator polynomial. That is, the bit probability is determined by the generator polynomial in the case of the ECC. The bit probabilities depending on the bit index are calculated based on the codeword for the cases of BCH and RS codes. However, the probability calculation includes the effect of the generator polynomial. The generator polynomial can be regarded as a linear combination of the input data bits. Thus, the ECC-encoded data bits are the output of the linear summation of the input bits completed by the generator polynomial.  
As shown in Figure 11, depending on the ECCs, the corresponding statistical eye diagram has different probability distributions [15]. The difference between the BCH and RS codes is the unit of encoding and decoding. The BCH code encodes the data bits in a bit-wise fashion. However, the RS code encodes the data bits in a symbol-wise fashion. The symbol-wise encoding means that the errors are corrected for the given symbol. Thus, it affects the bit probability distribution as a result of the ECC. The BCH and RS codes improve BER performance as a result of the decoding. The encoded data also improve the SI performance in the case of an RS code. As can be seen in Figure 11b, the RS-encoded statistical eye diagram has a higher probability for ZERO. This means that the number of ZEROs is much larger than the number of ONEs. These ZERO-biased bits provide additional advantages: mitigated crosstalk noise and suppressed radiated electromagnetic interference (EMI) noise. The decreased number of rising and falling responses corresponds to a smaller number of bit transitions as discussed in the case of TMDS encoding. Likewise, each bit transition induces high-frequency radiation. Thus, the RS code also suppresses the radiated EMI during data transmission.