A method for recovering a signal by measuring the signal to produce a plurality of compressive sensing measurements, discarding saturated measurements from the plurality of compressive sensing measurements and reconstructing the signal from remaining measurements from the plurality of compressive sensing measurements. Alternatively, a method for recovering a signal comprising the steps of measuring a signal to produce a plurality of compressive sensing measurements, identifying saturated measurements in the plurality of compressive sensing measurements and reconstructing the signal from the plurality of compressive sensing measurements, wherein the recovered signal is constrained such that magnitudes of values corresponding to the identified saturated measurements are greater than a predetermined value.