This chapter presents the foundational theory underlying the parametric characteristic analysis method, which serves as the theoretical cornerstone of the nonlinear Characteristic Output Spectrum (nCOS) function approach. Specifically, it addresses the formulation of accurate parametric structures for frequency response functions, enabling the explicit representation of nonlinear system behavior in terms of critical design parameters. The parametric characteristic framework is built upon a rigorously defined operator and its associated mathematical properties. This formulation facilitates a series of theoretical developments, culminating in the establishment of the nCOS function. The method has demonstrated broad applicability in various engineering domains, including vibration control, energy harvesting, and fault diagnosis.