## 1 Introduction
### Step Responses Derivation  
Similar to other statistical methods, output responses and occurrence probability of the responses must be derived. In this subsection, a method to derive the output responses is explained. To derive output responses, design parameters affecting output responses and variables should be defined. In Figure 2, an equivalent circuit model with considerable parameters for deriving output responses are shown. The hierarchical power distribution network (PDN), which consists of on-chip PDN, package/interposer PDN, printed circuit board (PCB) PDN, and decoupling capacitors in each PDN, must be considered. To accurately reflect the ground noise, the PDN must be decomposed into the power and ground interconnects instead of the S-parameter model. If the PDN is modeled based on the S-parameter model, such as the touchstone file, ground noise will be added to the power noise, which results in exaggerated power noise and suppressed ground noise [22; 23; 24]. However, high-speed interconnects can be modeled using S-parameter models or equivalent circuit models. These parameters should be carefully chosen or extracted depending on a target system to be analyzed, since these parameters determine overall output response characteristics. This equivalent circuit can be mathematically modeled into simultaneous second order differential equations [20]. In such case, the final eye-diagram can be estimated solely using a mathematic tool such as MATLAB. However, mathematical modeling itself requires an insight, and it is also time consuming. In this work, the transient simulator HSPICE is used for the equivalent circuit. Still, only responses will be derived, so it requires a much smaller computational resource and time compared to a full eye-diagram simulation. The output response characteristics and occurrence are determined by the SSO buffer states. Agressor buffers can be in four different states. Therefore, the four different states of aggressor SSO buffer are set as variable to derive output responses and occurrence probabilities. The SSO combinations are determined by the four variables, which are the number of SSO buffers in pull-up/down transitions (\(N_{01}/N_{10}\)) and the number of SSO buffers remaining in steady states logic 'zero' (\(N_{00}\)), and logic 'one' (\(N_{11}\)), as shown in Figure 2 with red letters. If these buffers are randomly operating (un-correlated to each other), it is not coded. If the occurrence probability of each buffer is correlated somehow, it is coded and in this case, bit patterns must be considered. In the following subsection, the occurrence probabilities of the SSO buffers for random cases and coded cases are derived. For step-input response, 'one-to-zero' for pull-up, 'zero-to-one' for pull-down, 'zero-to-zero' for steady state 'one', and 'one-to-one' for steady state 'zero' are injected. For each step-input response to the victim buffer, the state of the SSO buffers are varied. If the total number of buffers is set as \(N_{T}\), there are \(4^{(N_{T}-1)}\) SSO combinations for each step-response. However, the SSO buffers remaining in steady states have minimal impacts on the output responses of the victim buffer. Therefore, output responses for cases when \(N_{01}+N_{10}\leq N_{T}-1\) are considered for more efficient derivation. By considering the above condition, a considerable SSO combination number reduces from \(4^{(N_{T}-1)}\) to \(N_{T}(N_{T}+1)/2\). In Figure 3, derivation of output responses is graphically explained. For all cases, step responses are derived and plotted. As can be seen from Figure 3, if all buffers are in the same pull-up or pull-down transition, response with a large fluctuation and slower slew-rate is derived due to large power/ground noise and SSO noise. With the same number of pull-up and pull-down buffers, power and ground currents are canceled out, resulting in reduced fluctuation with a faster slew-rate. The number of pull-up or pull-down transitions increases, and larger power/ground noise with slower slew-rate are determined.