A method and system is presented for sampling analog signals in a manner that avoids the effects of signal clipping due to a limited dynamic range. A method and device for sampling an analog input using multiple gains, or gain mask, is described. By using different gains during different time quanta, a subset of the sampled points may effectively be attenuated before being sampled and converted to a digital representation. If clipping occurs during the sampling process, the true values of the clipped samples may be interpolated using the amplitudes of the non-clipped samples, which may not have been attenuated. Such interpolation may include constructing and/or solving a constraint optimization problem using linear programming. In one embodiment, such a problem may be constructed and/or solved by using sign information from the clipped samples and/or by imposing a sparsity assumption on the signals during the reconstruction process.