Efficient DDD-based Term Generation Algorithm for Analog Circuit Behavioral Modeling Sheldon X.-D. Tan C.-J. Richard Shi Department of Electrical Engineering Department of Electrical Engineering University of California, Riverside, CA 92521, USA University of Washington, Seattle, WA 98195, USA stan@ee.ucr.edu shi@ee.washington.edu Abstract— An efficient approach to generating symbolic prod- uct terms for behavioral modeling of large linear analog circuits is presented. The approach is based on a compact determinant decision diagram (DDD) representation of transfer functions and characteristics of analog circuits. The new algorithm is based on the concept that a dominant term in a DDD graph can be found by searching the shortest path in the graph. But instead of travers- ing a whole DDD graph each time, we show that a shortest path can be found by just updating a small number of the newly added vertices after the first shortest path is found. Experimental results indicate that the new symbolic term generation algorithm out- performs both pure shortest path based algorithm and dynamic programming based algorithm, which is the fastest symbolic term generation algorithm published so far. I. I NTRODUCTION Behavioral modeling aims at generating compact and sim- ulation ready models for analog circuit blocks that capture the circuit characteristics of interests. Behavioral models can be used to speed up full system design analysis and verifica- tion. Behavioral modeling is a critical technique for emerging system-on-a-chip (SoC) designs as efficient implementations of analog building blocks in SoC systems becomes increas- ingly important. One way to derive behavioral models of ana- log modules is by means of symbolic analysis. As illustrated in [3], simple yet accurate symbolic expressions can also be interpretable by analog designers to gain the insight into cir- cuit behavior, performance and stability, and are important for many applications in circuit design such as transistor sizing and optimization, topology selection, sensitivity analysis, fault simulation, testability analysis and yield enhancement [4]. Research on symbolic analysis can date back to 1960s [6]. Recently, various schemes to drive approximate symbolic ex- pressions have been developed. Approximation can be carried out after generation of all the symbolic product terms [3, 11, 16]; or during generation [2, 15, 17] and even before genera- tion [5, 17]. Recently Tan and Shi proposed an efficient DDD graph based method of deriving simple yet accurate symbolic ex- pressions for behavioral modeling of linear(ized) analog cir- cuits [12]. Their DDD-based approximation method has both the reliability in the approximation-after-generation meth- ods [3, 11, 16], and the capability in approximation-before [5, 17]/during-generation [2, 15, 17] methods for analyzing large analog circuits. In this paper, we consider how to obtain the dominant prod- uct terms from exact or simplified DDD graphs representing circuit characteristics. An efficient algorithm was proposed in [12] where finding a dominant term is transformed into searching the shortest path in a DDD graph. Once the short- est path is found, it can be subtracted from the DDD graph by simple DDD graph operations. The next dominant term can be found on the resulting DDD graph in the same way. Re- cently, Verhaegen and Gielen presented another DDD-based dominant-term generation algorithm, which is based on dy- namic programming concept [14]. It is shown in [13] that the dynamic programming based algorithm is faster than the short- est path based algorithm in general but at cost of more memory use. In this paper, we present a more efficient shortest path based algorithm for dominant term generation. The success of the new algorithm is based on the observation that if the source vertex in a DDD graph are properly defined, a shortest path can be found by just updating a small number of the newly added vertices after the first shortest path is found. Experimental re- sults show that our incremental shortest path based algorithm outperforms both the pure shortest path based algorithm and the dynamic programming based algorithm for different types of analog circuits. This paper is organized as follows. Section II reviews the concepts of DDDs and s-expanded DDDs. Section III presents new incremental shortest path based algorithm. We also briefly review our implementation of the dynamic programming based algorithm based on DDD graphs. Experimental results are de- scribed in Section IV. Section V concludes the paper. II. DDDS AND s-EXPANDED COEFFICIENT DDDS In this section, we provide a brief overview of the notion of determinant decision diagrams [8]. We review how a s- expanded DDD can be used to represent the symbolic coef- ficients of a s polynomial.