Resonance demodulation technology has been widely used for its advantages on magnification and selection of fault information. However, the selection of filter parameters is affected by the subjectivity, and the analysis results are easily affected by the noisy signals under actual working conditions. To solve this problem, a method combining relative entropy and correlation coefficient is proposed to analyze and select Intrinsic Scale Components obtained by Local Characteristic-scale Decomposition, remove false components, and to some extent, denoise. Then, after reconstructing the remaining components, we use spectral kurtosis analysis to find out the most easily detected frequency bands of fault features and analyze them by resonance demodulation, aiming to diagnose gearbox failures. Finally, we compare this method with that based on the empirical mode decomposition. The research shows that the method proposed in this paper can effectively diagnose the gearbox fault and is better than the empirical mode decomposition.