Compressed and Quantized Correlation Estimators
Résumé
In passive monitoring using sensor networks, low en- ergy 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 im- portant problem in this context. Here we study the estimation of correlation functions between sensors using compressed acquisi- tion 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 ≥ M can be more advantageous than estima- tion 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 practi- cal situations, compressed and quantized estimators behave suffi- ciently correctly to be useful in, for example, time-delay estimation and model estimation.