# Quantization in Compressive Sensing: A Signal Processing Approach

## TL;DR

This paper investigates how finite-length registers and quantization affect the accuracy of compressive sensing signal reconstruction, providing a mathematical model and validating it with numerical examples.

## Contribution

It introduces a unified model for quantization noise in CS and derives an exact formula for expected reconstruction error, considering signal nonsparsity and quantization effects.

## Key findings

- Quantization impacts CS reconstruction accuracy significantly.
- Derived an exact formula for expected error in quantized CS.
- Validated the theory with numerical examples including noise and approximate sparsity.

## Abstract

Influence of the finite-length registers and quantization effects on the reconstruction of sparse and approximately sparse signals 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 involve the finite-length registers and quantization of the measurements. An analysis of the effects related to the measurements quantization with an arbitrary number of bits is the topic of this paper. A unified mathematical model for the analysis of the quantization noise and the signal nonsparsity on the CS reconstruction is presented. An exact formula for the expected energy of error in the CS-based reconstructed signal is derived. The theory is validated through various numerical examples with quantized measurements, including the cases of approximately sparse signals, noise folding, and floating-point arithmetics.