Frequency-Scaled Semisymbolic Analysis

Semisymbolic analysis is one of the most valuable procedures in the automated design of circuits because it provides the poles and zeros of a transfer function. The algorithm itself consists of formulating a generalized eigenvalue problem, which is subsequently transformed into a standard one to be finally solved by the modification of a QR or QZ algorithm. Semisymbolic analysis itself has been known for a very long time, and its problems with numerical accuracy have also been known for a very long time. Since the advent of very fast circuits, this problem has heightened due to the presence of extremely small capacitances and inductances, which make the values in the matrices differ by a huge number of orders of magnitude. One option to solve this problem of numerical instability is to use more accurate arithmetics such as 128-bit numbers (and our previous works have assessed this possibility). In this article, however, we propose a more sophisticated procedure based on frequency scaling, which naturally balances the magnitudes of the matrix elements. The proposed algorithm is thoroughly verified in this study by a number of control analyses demonstrating that the use of the frequency scaling allows for accurate results to be achieved even using standard 64-bit arithmetic. Moreover, the article also shows that the implementation of frequency scaling into the subroutines for semisymbolic analysis is very easy. The overall efficiency of the newly developed algorithm is summarized in a final table that clearly shows that the number of obtained accurate and almost accurate poles and zeros increased significantly after the implementation of the suggested frequency scaling.