Frequency-Scaled Semisymbolic Analysis

Semisymbolic analysis is one of the most valuable procedures in an automated design of circuits because it provides poles and zeros of a transfer function. The algorithm itself consists in formulating generalized eigenvalue problem, which is subsequently transformed into a standard one to be finally solved by a modification of QR or QZ algorithm. Although the semisymbolic analysis itself is known for a very long time, its problems with numerical accuracy are known for a very long time as well. Since the advent of very fast circuits, this problem has heightened due to the presence of extremely small capacitances and inductances, which makes the values in the matrixes differing in 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 a frequency scaling, which naturally balances the magnitudes of the matrix elements. The proposed algorithm is thoroughly verified in the article by a number of control analyzes demonstrating that the use of the frequency scaling allows to achieve accurate results even by standard 64-bit arithmetic. Moreover, the article also shows that the implementation of the frequency scaling into the subroutines for the semisymbolic analysis is really very easy.