The analysis and design of nonlinear systems remain a persistent challenge in the literature, while also being critically important across a wide range of engineering applications—including dynamic modeling, response prediction, performance enhancement, and fault diagnosis. Over the years, numerous methodologies have been proposed to address these challenges. Among them, the harmonic balance method stands out as a widely adopted and extensively studied technique for nonlinear dynamic analysis, with diverse applications. A concise overview of such conventional methods is provided in the preceding chapter. In the following chapters, we focus on a frequency-domain approach grounded in Volterra series theory—namely, the nonlinear Characteristic Output Spectrum (nCOS) function method. The nCOS framework is formulated based on the concept of Generalized Frequency Response Functions (GFRFs), which serve as a natural extension of the classical transfer function paradigm used in linear system theory.This formulation enables a systematic and parametric representation of nonlinear system behavior in the frequency domain, offering enhanced analytical capabilities for both theoretical investigation and practical design. This chapter is to present how to calculate the GFRFs of a nonlinear system and thus serves to provide fundamentals of the nCOS framework.