Finite Range Scalar Quantization for Compressive Sensing

Analog-to-digital conversion comprises of two fundamental discretization steps: sampling and quantization. Recent results in compressive sensing (CS) have overhauled the conventional wisdom related to the sampling step, by demonstrating that sparse or compressible signals can be sampled at rates much closer to their sparsity rate, rather than their bandwidth. This work further overhauls the conventional wisdom related to the quantization step by demonstrating that quantizer overflow can be treated differently in CS and by exploiting the tradeoff between quantization error and overflow. We demonstrate that contrary to classical approaches that avoid quantizer overflow, a better finite-range scalar quantization strategy for CS is to amplify the signal such that the finite range quantizer overflows at a pre-determined rate, and subsequently reject the overflowed measurements from the reconstruction. Our results further suggest a simple and effective automatic gain control strategy which uses feedback from the saturation rate to control the signal gain.