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We use recently developed convex programming techniques to reconstruct arbitrary sparse signals observed through projections onto a small-dimensional space in background noise in order to estimate and remove impulsive noise in an OFDM system. We develop deterministic construction of projection matrices that provably guarantee reconstruction with high probability. Finally, we compare the achievable rate using our novel method with some simple capacity lower and upper bounds and with the recently obtained capacity of the Gaussian erasure channel. For practical impulse probability the proposed scheme appears to be competitive. This scheme may find some application in DSL and powerline communications, where transmission is typically affected by intersymbol interference, Gaussian noise and impulsive noise.