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An eye diagram, a critical metric in signal integrity analysis for high-speed interconnects such as packages, interposer, and printed circuit boards (PCBs), is generated by superposition of the received waveform. Obtaining an eye diagram is time-consuming, thus signal integrity analysis is inefficient. This article reviews that have been proposed to overcome this limitation. The statistical eye diagram provides a probability distribution depending on a sampling time and voltage, therefore it can be expanded to other metrics, such as the bit-error rate and shmoo plot. This article introduces previous research on statistical eye diagrams applied to complementary metal-oxide-semiconductors (CMOSs), noise, and high-speed systems. The methods applied to CMOSs include asymmetry between the P/NMOS transistors and the nonlinearity of the CMOS. The methods applied to noise include signal and power noise. The methods applied to high-speed systems include equalizers, signaling, encoding, linear feedback shift register, and error correction code.

This article proposes a statistical shmoo for a Decision Feedback Equalizer (DFE). A statistical shmoo shows a Bit-Error Rate (BER) distribution of a system depending on conditions such as a threshold voltage and sampling time. The previous statistical eye diagram can be converted to a statistical shmoo except for the case of a DFE. In the shmoo, rows and columns represent different threshold voltages and sampling times, respectively. Thus, the DFE operation also depends on each row and column. Furthermore, a DFE may have error propagations because of substantial noise. Therefore, this article proposes a statistical shmoo for an ideal and a nonideal DFE. The ideal DFE always makes a correct bit decision, whereas the nonideal DFE may make a wrong bit decision depending on the received waveform. For verification, the obtained statistical shmoo is compared to the transient simulation with their BER values. The lowest BER value achieved by the proposed method is 1E-51 with a calculation time of 862 seconds. While the transient simulation achieved the BER of 8.26E-8 with a simulation time of 2,000,000 seconds. In conclusion, the proposed method provides an accurate statistical shmoo in efficient manner.

This article proposes an accelerated statistical eye diagram estimation method for efficient signal integrity (SI) analysis. An eye diagram is a critical metric in the SI analysis, however obtaining an eye diagram is time-consuming in simulation. The required time might be a few weeks depending on the complexity. Because the eye diagram is obtained by superposition of the received waveforms. Statistical eye diagram estimation methods were proposed to make the eye diagram acquisition efficient, however, it still has a limited improvement due to a large number of convolution operations. To address this limitation, the proposed method includes two approaches: ( $i$ ) an increased voltage step in the amplitude PDF with a compensation factor and (ii) skippable and replaceable convolution operations. The efficiency of the proposed method is verified by comparing the derived result with that of a conventional transient simulation. The previous estimation method requires 1,065 s while approaches ( $i$ ) and (ii) reduce the required time to 88 and 510 s, respectively. When both approaches are applied, the required time is reduced to 17 s in comparison to the simulation time of 259,200 s required for a transient simulation. Approach (ii) reduces the number of convolution operations from 768,000 to 103,000. Therefore, the proposed method successfully reduces the required simulation time and the required number of convolutions to obtain a statistical eye diagram.