Quantization in Compressive Sensing: A Signal Processing Approach
The influence of finite-length registers and the corresponding quantization effects on the reconstruction of sparse and approximately sparse signals from a reduced set of measurements is analyzed in this paper. For the nonquantized measurements, the compressive sensing (CS) framework provides highly accurate reconstruction algorithms that produce negligible errors when the reconstruction conditions are met. However, hardware implementations of signal processing algorithms inevitably involve finite-length registers and quantization of the measurements. A detailed analysis of the effects related to the measurement quantization, with an arbitrary number of bits, is provided in this paper. A unified novel mathematical model to characterize the influence of the quantization noise and the signal nonsparsity on the CS reconstruction is introduced. Using this model, an exact formula for the expected error energy in the CS-based reconstructed signal is derived, while in the literature its bounds have been reported only. The theory is validated through various numerical examples with quantized measurements, involving scenarios with approximately sparse signals, noise folding effect, and floating-point arithmetics.