Distributed adaptive quantization for wireless sensor networks: From delta modulation to maximum likelihood

We consider distributed parameter estimation using quantized observations in wireless sensor networks (WSNs) where, due to bandwidth constraint, each sensor quantizes its local observation into one bit of information. A conventional fixed quantization (FQ) approach, which employs a fixed threshold for all sensors, incurs an estimation error growing exponentially with the difference between the threshold and the unknown parameter to be estimated. To address this difficulty, we propose a distributed adaptive quantization (AQ) approach, which, with sensors sequentially broadcasting their quantized data, allows each sensor to adaptively adjust its quantization threshold. Three AQ schemes are presented: 1) AQ-FS that involves distributed delta modulation (DM) with a fixed stepsize, 2) AQ-VS that employs DM with a variable stepsize, and 3) AQ-ML that adjusts the threshold through a maximum likelihood (ML) estimation process. The ML estimators associated with the three AQ schemes are developed and their corresponding Cramér-Rao bounds (CRBs) are analyzed. We show that our 1-bit AQ approach is asymptotically optimum, yielding an asymptotic CRB that is only π2 times that of the clairvoyant sample-mean estimator using unquantized observations.