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Impulsive noise is the bottleneck that determines the maximum length of the DSL. Impulsive noise seldom occurs in DSL but when it occurs, it is very destructive and results in dropping the affected DSL symbols at the receiver as they cannot be recovered. By considering impulsive noise a sparse vector, recently developed sparse reconstruction algorithms can be utilized to combat it. We propose an algorithm that utilizes the null carriers for the impulsive noise estimation and cancellation. Specifically, we use compressive sampling for a coarse estimate of the impulse position, an a priori information based MAP metric for its refinement, followed by MMSE estimation for estimating the impulse amplitudes. We also present a comparison of the achievable rate in DSL using our algorithm and recently developed algorithms for sparse signal reconstruction.