Efficient Time-Domain Vector Fitting for Broad- band Interconnect Modelling Se-Jung Moon #1 , Xiaoning Ye #2 , Andreas Cangellaris *3 # Intel Hillsboro OR USA 1 se-jung.moon@intel.com 2 xiaoning.ye@intel.com * ECE Department, University of Illinois at Urbana Champaign 1406 W. Green St.Urbana, IL 61801, USA 3 cangella@illinois.edu Abstract—The time-domain vector fitting (TDVF) [1] was proven to be an extrapolation method which provides about a ten-fold lengthening of a recorded time-domain response and reducing Gribbs phenomenon due to the limited length of the response and its abrupt discontinuity. TDVF is the counterpart of the vector fitting (VF) [2] in the time domain (TD), but it is not popular as VF due to the fact that a technique for estimating the order of the rational function fit for TDVF is not available even though the accuracy of the model built by TDVF strongly depends on the choice of the order of the rational function fit. This paper introduces a new methodology toward an efficient and robust whole channel simulation using the TDVF algorithm which is equipped with a sound estimate for the order of the rational function fit. This method significantly reduces the computational resource, while enhances the model accuracy. As a validation study, TDVF equipped with the order estimation method was applied for modelling a socket component in multi- Gbps interconnects. I. INTRODUCTION As operating frequencies in very large-scale integration (VLSI) expand well into the microwave realm, switching speeds get faster and interconnect packaging density increases, the electromagnetic attributes of the chip-to-motherboard interface contribute to significant degradation of the transmitted signals, manifested in terms of attenuation, reflection, radiation, delay, and distortion. This degradation must be accurately captured in the predictive modelling and simulation in order to provide for an accurate circuit simulation-aided design and optimization of the channel [3]. It is common practice to use rational function fitting techniques to generate broadband circuit simulation- compatible transfer functions for high-speed connectors and interfaces, making use of response data obtained either through measurement or through electromagnetic field solvers. Among the various approaches for generating a rational function approximations, the time-domain vector fitting TDVF method [1] has been shown to be an extrapolation method capable of providing as much as a ten-fold lengthening of the recorded time-domain response of a passive electromagnetic device. However, TDVF has not been prevalently used. Thus, even for cases where measured or numerically obtained time-domain (TD) data are available for macromodeling, the customary practice is to first convert the time-domain data into frequency domain data using the Fourier transformation, and then apply a frequency-domain rational function fitting process [4], such as Vector Fitting VF [2], to fit the transfer function. The main reason for this is that a technique for estimating the order of the rational function fit using for TDVF is not available. This is an inhibiting factor given that the success of the TDVF in generating an accurate fit is strongly depends on the sound choice of the order of the rational function fit. Recently we introduced an efficient order estimation method for TDVF [5] based on the 2WT theorem [6,7]. In this paper, we demonstrate the effectiveness and accuracy of TDVF as an interpolation/extrapolation algorithm when combined with a reliable rule for the prediction of the order of the interpolation. We do this in the context of the element-by- element modelling of the scattering parameter matrix of a socket component in multi-Gbps interconnects [8]. Once an accurate interpolation has been generated, a SPICE- compatible equivalent circuit is readily synthesized for subsequent use in system-level circuit simulation studies [9]. II. INTERCONNECT MODELLING A. Time-Domain Vector Fitting For simplicity and without loss of generality, a single-input single-output (SISO) passive system is considered since each element in the multi-port network can be treated as SISO when the element-by-element fitting method is used [8]. The purpose of TDVF is to construct the rational function approximation () fit H s of the transfer function, in this case, a scattering parameter (S-parameter), using the input () xt and the impulse response ( ). yt The rational function fit of () fit H s can be presented as 1 () N n fit n n r H s d s q = ≈ + − ∑ (1) 978-1-4244-6307-7/10/$26.00 ©2010 IEEE 563