Determining Input-Output Relationships of Nonlinear Systems by Inversions

The relation between input and output of many linear systems and circuits is ordinarily found by repeated use of functional inversion, i.e., the process where, given
\bar{y} = Z\bar{x}
one finds
\bar{x} = (1 /Z)\bar{y}
. Many nonlinear circuits and systems can be analyzed using functional inversion in exactly the same way. The methods for accomplishing this inversion however are different. Where the nonlinear relations are given as graphs, the inverse relations are shown by the same graphs. Where derivatives and integrals are involved, approximate inverses can be obtained using a generalization of the "reversion" method of Sims [11] and Pipes [12], or the iteration method. Practical examples given include the design of a nonlinear equalizer and the analysis of a feedback circuit, a tunnel diode network, and an automatic gain control circuit. The mathematical bases of the methods are discussed, along with their representation by flow graphs, and the insertion of initial conditions.