Cram\'er-Rao Bounds for Polynomial Signal Estimation using Sensors with AR(1) Drift

TL;DR

This paper derives approximate Cramér-Rao bounds for polynomial signal estimation in sensor networks affected by AR(1) drift, revealing how drift impacts estimation accuracy and providing insights for practical sensor network design.

Contribution

It introduces closed-form Fisher Information and Cramér-Rao bounds for signals with AR(1) drift, addressing a gap in understanding systematic errors in sensor estimation.

Findings

For <1, drift scales measurement noise asymptotically.

For =1, constant signals cannot be estimated consistently.

Results are applicable to multi-sensor and bandwidth-limited networks.

Abstract

We seek to characterize the estimation performance of a sensor network where the individual sensors exhibit the phenomenon of drift, i.e., a gradual change of the bias. Though estimation in the presence of random errors has been extensively studied in the literature, the loss of estimation performance due to systematic errors like drift have rarely been looked into. In this paper, we derive closed-form Fisher Information matrix and subsequently Cram\'er-Rao bounds (upto reasonable approximation) for the estimation accuracy of drift-corrupted signals. We assume a polynomial time-series as the representative signal and an autoregressive process model for the drift. When the Markov parameter for drift \rho<1, we show that the first-order effect of drift is asymptotically equivalent to scaling the measurement noise by an appropriate factor. For \rho=1, i.e., when the drift is non-stationary, we show that the constant part of a signal can only be estimated inconsistently (non-zero asymptotic variance). Practical usage of the results are demonstrated through the analysis of 1) networks with multiple sensors and 2) bandwidth limited networks communicating only quantized observations.