Normalized Coefficient Linear Combinations (NCLC): A Unifying Framework for Discrete-Time Filters, Control, and Signal Analysis

This paper presents the Normalized Coefficient Linear Combination (NCLC) as a unifying structural framework for analyzing systems in digital signal processing (DSP), discrete-time control, and numerical sequence analysis. While traditional design focuses on pole–zero placement, the NCLC perspective emphasizes the decomposition of signals into a global carrier function (scale) and a residual modulated by normalized coefficients. We formalize conditions under which this structure guarantees asymptotic preservation of the carrier. Furthermore, we demonstrate how coefficient normalization serves as a direct design tool in finite-order discrete-time LTI systems to simultaneously satisfy BIBO stability and prescribed steady-state gain constraints. The versatility of this framework is illustrated through three applications: a first-order digital filter, a parametric model correction scheme, and the signal analysis of prime number asymptotics viewed as a discrete sequence with arithmetic noise.