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Discrete-time filters represent a promising solution for pulse-processing in high-energy physics experiments due to their flexibility, reliability, and their capability to synthesize weighting functions with virtually any shape. One of the major concerns when designing one of these filters is to calculate the filter parameters that maximize the signal-to-noise ratio. The classic way to address this problem is to perform the noise analysis using a continuous-time domain approach based on the weighting function concept. However, when addressing the problem from an inadequate time domain, the analysis is not insightful and the resulting expressions are complex and difficult to use for design purposes. In this work, a mathematical framework for a design-oriented analysis of discrete-time filters in the discrete-time domain is presented. This analysis is based on treating the sampled noise as a discrete-time signal, which can be manipulated to obtain a closed-form expression for the front-end noise, suitable for computer automatic evaluation and filter optimization procedures. An example of the optimum filter formulation and computation is presented, in addition to several conclusions about optimum digital filtering.