As an improvement version of empirical mode decomposition (EMD), local characteristic-scale decomposition (LCD) also has the problem of mode mixing. The noise-assisted methods, such as ensemble EMD (EEMD) and complete EEMD (CEEMD), are effective ways to restrain the mode mixing. However, pseudo-components are easily generated by them and the obtained components sometimes do not meet the definition of IMF. A novel signal processing method called complete ensemble LCD (CELCD) is proposed to resolve mode mixing involved in LCD. Also, the comparisons between CELCD and CEEMD are studied by analyzing simulation signal, and the results indicate that CELCD could effectively restrain the mode mixing of LCD and generate more precise components. The rotor rubbing fault diagnosis verifies the validity of the proposed method.