The impulsive noise suppression problem for single-carrier power line communication based on turbo equalisation is investigated. In this work, the sparse property of the impulsive noise in time domain is utilized to actively suppress the noise. Based on this idea, an optimization framework is developed to estimate the impulsive noise, which is effectively solved by the alternating direction method of multipliers (ADMM). For the power line communication, due to the low complexity and low peak-to-average power ratio (PAPR), the single-carrier frequency-domain equalization is used in this work. Numerical studies demonstrate the superior performance of the proposed approach.