In general, during the whole life cycle operation of mechanical system, its corresponding condition signal often presents multi-component polymorphic-oscillatory characteristic and is accompanied with strong interference noise. In order to identify system operating status, how to achieve polymorphic signal decomposition is an unavoidable focus. Weak signal features corrupted by heavy background noise can be effectively extracted through sparse decomposition. In order to solve the problems of classical sparse decomposition method, such as the lack of signal fidelity, the local optimal solution caused by the non-convex objective function, and the poor universality of the model, a novel multi-source sparse optimization objective function with convexity is constructed based on the generalized mini-max concave penalty function. Then the sparse coefficients of unilateral attenuation transient component, bilateral attenuation transient component and harmonic component are calculated respectively based on Laplace wavelet dictionary, Morlet wavelet dictionary and DFT dictionary using forward backward splitting algorithm. Ultimately, each distinct component can be extracted based on these sparse coefficients. Comparison with the classical resonance sparse signal decomposition (RSSD) based on L1-norm, signal adaptive decomposition and spectral kurtosis show that the proposed method can accurately preserve the amplitude of morphological components under the low SNR premise. Experimental case infers that the proposed method compared with double tree complex wavelet (DTCW) possesses potential value of application on mechanical system fault detection without the prior knowledge of specific number of faults.