Statically constrained non-linear models with application to IC buffers C. Diouf 1 , M. Telescu 1 , N. Tanguy 1 , P. Cloastre 1 , I.S. Stievano 2 , F.G. Canavero 2 1 Université Européenne de Bretagne, France. Université de Brest ; CNRS, UMR 3192 Lab-STICC 2 Politecnico di Torino, Italy Reference contact: mihai.telescu@univ-brest.fr Abstract Volterra series are well known approaches to modeling non-linear systems. In recent works a properly weighted combination of Volterra models have been used to successfully mimic the behavior of the output port of an integrated circuit buffer. The current paper focuses on a novel mechanism for controlling the static characteristic of such models. A commercial driver example is used to illustrate the efficiency of the technique to guarantee accurate static levels during time-domain simulations. Introduction The modeling of digital Integrated Circuits (IC) input and output buffers is a vital stage in the assessment of signal integrity and electromagnetic compatibility in high-end digital systems. Buffers both drivers and receivers act as non-linear terminals of the interconnect line networks and their transient behavior may have unwanted effects on the proper functioning of the system. In recent years, researchers have concentrated their attention on the development of behavioral models that seek to mimic the device’s input-output behavior of ten proves more efficient from a computational point of view than attempting to model its internal physics. Different approaches exist ranging from neural networks [1] to more common two-piece models [2-3] exploiting the two-state nature of the device. In a previous paper [4] some of the authors showed that two properly weighted weakly non- linear Volterra models could successfully be used to mimic the output behavior of drivers. The strength of this approach resides in a greater degree of generality allowing for greater adaptability. In this paper the authors focus on the static characteristic of Volterra models and the way in which they may be forced to fit a certain form. This is particularly interesting in the field of buffer modeling where the static behavior of the original system needs to be accurately reproduced. Volterra and Volterra-Laguerre models Volterra series are a well known input-output representation of non-linear systems and have found in the last half-century a variety of applications in different fields of science. They have proved to be a valuable and reliable tool in nonlinear system identification and allow a straightforward generalization to multivariable systems. The general discrete time expression of a Volterra series is given by 1 0 0 1 2 1 0 0 2 1 2 1 2 0 1 1 1 ) ( ) ,..., , ( ... ... ) ( ) ( ) , ( ) ( ) ( ) ( 1 2 1 m k k m l l m m k k k j m k k x k k k h k k x k k x k k h k k x k h k y (1) with k being the discrete time, y the output variable, x the input and h m being referred to as the m-th order kernel. One may intuitively see Volterra series as a generalization of a linear systems impulse response function and it is quite obvious that for m = 1 expression (1) is reduced to the classical impulse response function of a linear system. In practice direct identification of the kernels, for an unknown nonlinear system, may prove cumbersome. A common approach is to use a projection on an orthogonal basis. The discrete Laguerre basis has been chosen to illustrate the approach presented in this paper, although the mathematical derivations may be extended to other representations. The discrete Laguerre functions k i may be defined by their z transform as follows ,... 1 , 0 , 1 1 ) ( 2 i a z az a z z a z i i (2) Laguerre functions form a complete set of orthonormal functions in 2 , then each Volterra kernel can be expressed with a multidimensional Laguerre series as 0 0 2 1 , 2 1 2 0 1 1 1 1 2 2 1 2 1 1 1 1 ) ( ) ( ) , ( ) ( ) ( i i i i i i i i i k k C k k h k C k h (3) Let k i denote the response of the ith Laguerre filter with transfer function given by (2) to the input k x , i.e. 0 l k l l i i i k k x k k x k k . (4) It follows that equation (1) can be recast as