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Impulsive noise is the bottleneck that limits the distance at which DSL communications can take place. 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 guard band null carriers for the impulsive noise estimation and cancellation. Instead of relying on ℓ1 minimization as done in some popular general-purpose compressive sensing (CS) schemes, the proposed method exploits the structure present in the problem and the available a priori information jointly for sparse signal recovery. The computational complexity of the proposed algorithm is very low as compared to the sparse reconstruction algorithms based on ℓ1 minimization. A performance comparison of the proposed method with other techniques, including ℓ1 minimization and another recently developed scheme for sparse signal recovery, is provided in terms of achievable rates for a DSL line with impulse noise estimation and cancellation.