We explore and exploit a heretofore relatively unexplored hallmark of compressive sensing (CS), the fact that certain CS measurement systems are democratic, which means that each measurement carries roughly the same amount of information about the signal being acquired. Using this property, we re-think how to quantize the compressive measurements. In Shannon-Nyquist sampling, we scale down the analog signal amplitude (and therefore increase the quantization error) to avoid the gross saturation errors. In stark contrast, we demonstrate a CS system achieves the best performance when we operate at a significantly nonzero saturation rate. We develop two methods to recover signals from saturated CS measurements. The first directly exploits the democracy property by simply discarding the saturated measurements. The second integrates saturated measurements as constraints into standard linear programming and greedy recovery techniques. Finally, we develop a simple automatic gain control system that uses the saturation rate to optimize the input gain
