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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 to detect and estimate the local spectral irregular structure, which carries important information on the frequency locations and power spectral densities of the sensed subbands. In this context, a wideband spectrum sensing techniques was developed based on an analog decision function to multi-scale wavelet product. The proposed sensing techniques provide an effective sensing framework to identify and locate spectrum holes in the signal spectrum.