Compact representation and efficient generation of s-expanded symbolic network functions for computer-aided analog circuit design

Abstract

Abstract-A graph-based approach is presented for the generation of exact symbolic network functions in the form of rational polynomials of the complex frequency variable for analog integrated circuits. The approach employs determinant decision diagrams (DDDs) to represent the determinant of a circuit matrix and its cofactors. A notion of multiroot DDDs is introduced, where each root represents a symbolic expression for an individual coefficient of the powers of in the numerator and denominator of a network function, and multiple roots share their common subgraphs. A DDD-based algorithm is presented for generating -expanded network functions. We prove theoretically and validate experimentally that the algorithm constructs in ( DDD ) time an -expanded DDD with no more than DDD vertices, where is the degree of the denominator polynomial, is the maximum number of devices that connect to a circuit node, and DDD is the number of DDD vertices representing the circuit-matrix determinant. For a practical circuit, DDD is often many orders-ofmagnitude less than the number of product terms. In contrast, previous approaches require the time and space complexities proportional to the number of product terms, which grows exponentially with the size of a circuit. Experimental results have demonstrated that the new approach can produce exact -expanded-symbolic network functions for A741 operational amplifiers in several CPU seconds on an UltraSparc-I workstation. The expressive power of multiroot -expanded DDDs is so remarkable that in one instance, over 10 35 symbolic product terms have been represented by a multiroot DDD with less than 17 K vertices. The compactness of DDDs is further demonstrated in the context of symbolic noise evaluation, where potentially many transfer functions, each being used for a noise source in the circuit, can be represented by a single DDD with the size comparable to that for a few transfer functions. This provides a powerful tool for solving many symbolic analysis problems such as deriving interpretable symbolic expressions, dominant pole/zero estimation, and analog testability analysis. We have also demonstrated that repetitive numerical evaluation with the derived -expanded symbolic expressions for frequency-domain simulation and small-signal noise analysis can be much faster than SPICE-like simulators and the resulting expressions for a circuit block can be used as behavioral models for high-level simulation.

Efficient DDD-based term generation algorithm for analog circuit behavioral modeling

“…For each circuit, DC analysis is first carried out using SPICE and our program reads in small-signal element values from the SPICE output. The algorithms described in [8,10] are used to construct complex DDDs and s-expanded DDDs. Table I summarizes the comparison results in terms of CPU time for the three algorithms.…”

“…The s-expanded DDD can be constructed from the complex DDD in linear time in the size of the original complex DDD [9,10].…”

“…Their implementation, however, is based on the DDD structure that is slightly different from the one developed originally by Tan and Shi [8,10]. Inspired by their work, we implemented a DP based algorithm based on the original DDD structure.…”

An efficient algorithm for transient and distortion analysis of mildly nonlinear analog circuits

“…Also like BDDs, DDDs are very capable of representing huge number of symbolic terms from a determinant. Most importantly, it can derive the s-expanded polynomial of a determinant symbolically via s-expanded DDDs [7]. The recent hierarchical approach using DDD graphs can essentially derive transfer functions for almost arbitrary large networks [8], which makes the solving of linear networks in frequency domain much easy and efficient.…”

“…Our method is also based on the Volterra functional series. But instead of solving the Volterra circuits in time domain as done by traditional methods like SPICE3 or by the sampled-data method [6], we solve the Volterra circuits in frequency domain by using a graph-based symbolic analysis method [7,8]. Once frequency domain responses are obtained, transient responses can be obtained by fast numerical inverse Laplace transformation [1].…”