On Vector Fitting Methods in Signal/Power Integrity Applications Chi-Un Lei † , Yuanzhe Wang, Quan Chen and Ngai Wong Abstract—Vector Fitting (VF) has been applied to reformu- late traditional system identification techniques by introducing a partial-fraction basis to avoid ill-conditioned calculation in broadband system identifications. Because of the reliable and versatility of VF, many extensions and applications have been proposed, for example, the macromodeling of linear structures in signal/power integrity analyses. In this paper, we discuss the macromodeling framework and some main features in VF in terms of data, algorithms and models. Finally, an alternative P -norm approximation criterion is proposed to enhance the macromodeling process. Index Terms—Signal/Power Integrity, Vector Fitting, Macro- modeling, Tutorial, Approximation I. I NTRODUCTION Vector Fitting (VF) [1] is a numerical technique for sam- pled response-matching system identification (macromodel- ing), which involves iterative linear least-squares solves with a partial fraction basis. As opposed to other system identification techniques for broadband (from DC to GHz) system identi- fication, VF avoids ill-conditioned calculation, and therefore works in a more robust and efficient manner. Furthermore, its theoretically-simple and versatile framework can easily incooperate various constraints by introducing a variety of extensions for other areas. VF has also been used in modeling of different electrical systems [1], [2] and extended to differ- ent areas, for example, filter design [3]–[5], power network analysis [2], [6] and electromagnetic (EM) simulation [7]. The idea of VF was firstly introduced for transmission line transient modeling in [8]. The underlying idea of VF is to replace the approximated (or initialized) poles with an improved set of poles through implicit weighting (the pole relocation technique), which thereby improves the approxima- tion iteratively. VF approximates an underlying system to a new system using partial fraction basis with real or complex conjugate poles. A number of generalizations and extensions have been proposed for better VF performance and integration with various identification requirements [9]–[24]. VF has been thoroughly discussed in [25], [26]. Its basic implementation is available from [27], whereas its variants have been widely used in industrial electronic design workflows for signal integrity issues [28]–[30]. This paper acts as a tutorial on VF. We first give a brief introduction to the signal/power integrity issues (Section II) † Corresponding Author. C-U. Lei, Y. Wang, Q. Chen and N. Wong are with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong. Phone: ++852 +2859 2698 Fax: ++852 +2559 8738 Email: {culei, yzwang, quanchen, nwong}@eee.hku.hk Fig. 1. Common macromodeling flow in signal integrity analyses. and basic formulation of VF (Section III). Then we discuss the applications of VF in system identification (Sections IV, V and VI). Finally an alternative P -norm approximation cri- terion in VF is proposed for approximation enhancement (Section VII), which is verified through numerical examples (Section VIII). II. MACROMODELING:SYSTEM I DENTIFICATION PROBLEM IN SIGNAL/POWER I NTEGRITY With the increasing operational frequency and decreasing size of integrated circuits (ICs), high-frequency effects, such as signal delay and crosstalk, have become dominant factors lim- iting system performance in IC design. Accurate and efficient simulation is required to capture the high-frequency behavior of systems, so as to ensure consistent transmissions and reli- able ground (and power) distributions in high-speed electronic systems [2], [31]. A common simulation flow is shown in Fig. 1. The sampled structure responses can be obtained by ex- citing one input port at a time and computing or measuring the responses at the output ports (Response Characterization). By approximating the sampled frequency-dependent or time- dependent system response data, a macromodel is generated to replace the original large-order system by a smaller-order one with similar input-output relationship (Macromodeling). The macromodel is used to generate spectra and waveforms for signal integrity analysis or coupled with other circuit model blocks (e.g., logic devices) for global simulation (Simulation). Peripheral pre-processing and post-processing techniques are used to rectify the macromodel characteristics and enhance the simulation performance. Generally, for a single-port (one input port and one out- put port) system, macromodeling techniques intend to fit a linear-time invariant (LTI) system to the desired continuous- time frequency-sampled response H (s) at a set of calcu- lated/sampled points at the input and output ports. The model