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This paper presents a novel technique in spectrum sensing based on a new characterization of primary users signals in wideband communications. First, we have to remind that in cognitive radio networks, the very first task to be operated by a cognitive radio is sensing and identification of spectrum holes in the wireless environment. This paper summarizes the advances in the algebraic approach. Initial results have been already disseminated in few other conferences. This paper aims at finalizing and presenting the last results and the complete framework of the proposed technique based on algebraic spectrum discontinuities detection. The signal spectrum over a wide frequency band is decomposed into elementary building blocks of subbands that are well characterized by local irregularities in frequency. As a powerful mathematical tool for analyzing singularities and edges, the algebraic framework is employed …