Compressed and quantized correlation estimators

TL;DR

This paper investigates the estimation of correlation functions between sensors using compressed sampling and one-bit quantization, demonstrating advantages over traditional methods and analyzing their asymptotic performance.

Contribution

It introduces a novel approach for correlation estimation directly from compressed, quantized samples without signal reconstruction, including analysis for various random projection matrices.

Findings

Compressed estimators can outperform traditional methods when signals are far from white noise.

Random projection matrices like Gaussian, Bernoulli, and sparse matrices are effective for compression.

Empirical results show practical usefulness in time-delay and model estimation tasks.

Abstract

In passive monitoring using sensor networks, low energy supplies drastically constrain sensors in terms of calculation and communication abilities. Designing processing algorithms at the sensor level that take into account these constraints is an important problem in this context. We study here the estimation of correlation functions between sensors using compressed acquisition and one-bit-quantization. The estimation is achieved directly using compressed samples, without considering any reconstruction of the signals. We show that if the signals of interest are far from white noise, estimation of the correlation using $M$ compressed samples out of $N\geq M$ can be more advantageous than estimation of the correlation using $M$ consecutive samples. The analysis consists of studying the asymptotic performance of the estimators at a fixed compression rate. We provide the analysis when the compression is realized by a random projection matrix composed of independent and identically distributed entries. The framework includes widely used random projection matrices, such as Gaussian and Bernoulli matrices, and it also includes very sparse matrices. However, it does not include subsampling without replacement, for which a separate analysis is provided. When considering one-bit-quantization as well, the theoretical analysis is not tractable. However, empirical evidence allows the conclusion that in practical situations, compressed and quantized estimators behave sufficiently correctly to be useful in, for example, time-delay estimation and model estimation.