Interpretable online updated weights: Optimized square envelope spectrum for machine condition monitoring and fault diagnosis

•A convex optimization model integrated with cyclostationarity is established.•The maximum logarithmic likelihood estimation is used to solve the convex optimization model.•Interpretable weights of the model learned from monitoring signals are used as optimized square envelope spectrum (OSES).•Three-dimensional OSES is adaptively updated to relieve the requirement of historical data.•The proposed OSES and three-dimensional OSES are new tools for machine condition monitoring and fault diagnosis.