A new method for causality enforcement of DRAM package models using discrete hilbert transforms

Abstract

Causality verification and enforcement is an essential step in generating high speed electric package macromodels and it often accomplished in two steps: vector fitting measured system parameters into a rational function representation and then performing the Hilbert Transform integrations to check and if needed enforce causality. This procedure suffers from various approximation, truncation, and discretization errors. Besides, the measured or simulated system data are known only on the finite bandwidth, while the Hilbert Transform has to be computed on the infinite domain. To avoid these errors as well as inaccuracy of extrapolation of the transfer function to infinity, a new method is proposed to check and enforce causality by using raw bandlimited data that is extended periodically using a polynomial interpolation and computing the Hilbert Transform of the periodically extended data via accurate FFT.