Authors’ summary: The purpose of this paper is to present a new atomic decomposition for a dense class of weighted Hardy spaces \(H^p_w(\mathbb{R}^n)\) via the discrete Calderón-type reproducing formula and the weighted Littlewood-Paley-Stein theory, where \(w\in A_\infty\) is a Muckenhoupt’s weight and \(0<p<\infty\). Our results can recover and improve the known ones in the literature by avoiding using the maximal function characterization and the Calderón-Zygmund decomposition. Moreover, we give a new proof of the weighted Hardy spaces estimates for generalized Calderón-Zygmund operators in terms of the atomic decomposition and the vector-valued singular operator theory. Although the theory of \(H^p_w(\mathbb{R}^n)\) is well known, we give new and simpler proofs, which in turn are amenable to utilization in general and nonclassical settings.