# On Distributed Estimation in Hierarchical Power Constrained Wireless Sensor Networks

## TL;DR

This paper investigates optimal power allocation and signal fusion strategies in hierarchical wireless sensor networks to minimize estimation error under power constraints, providing theoretical bounds and practical schemes.

## Contribution

It introduces a comprehensive framework for optimizing training, transmission, and fusion weights in hierarchical sensor networks, including analysis of special cases and complexity tradeoffs.

## Key findings

- Derived the MSE and Bayesian CRB for the estimation problem.
- Proposed optimal power and fusion weight schemes to minimize MSE.
- Quantified performance gains and complexity tradeoffs of different schemes.

## Abstract

We consider distributed estimation of a random source in a hierarchical power constrained wireless sensor network. Sensors within each cluster send their measurements to a cluster head (CH). CHs optimally fuse the received signals and transmit to the fusion center (FC) over orthogonal fading channels. To enable channel estimation at the FC, CHs send pilots, prior to data transmission. We derive the mean square error (MSE) corresponding to the linear minimum mean square error (LMMSE) estimator of the source at the FC, and derive the Bayesian Cramer-Rao bound (CRB). Our goal is to find (i) the optimal training power, (ii) the optimal power that sensors in a cluster spend to transmit their amplified measurements to their CH, and (iii) the optimal weight vector employed by each CH for its linear signal fusion, such that the MSE is minimized, subject to a network power constraint. To untangle the performance gain that optimizing each set of these variables provide, we also analyze three special cases of the original problem, where in each special case, only two sets of variables are optimized across clusters. We define three factors that allow us to quantify the effectiveness of each power allocation scheme in achieving an MSE-power tradeoff that is close to that of the Bayesian CRB. Combining the information gained from the factors and Bayesian CRB with our computational complexity analysis provides the system designer with quantitative complexity-versus-MSE improvement tradeoffs offered by different power allocation schemes.