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 Cramer-Rao bounds (up to 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 ρ <; 1, we show that the first-order effect of drift is asymptotically equivalent to scaling the measurement noise by an appropriate factor. For ρ = 1, i.e., when the drift is nonstationary, 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.
A related area of work is the study of systematic-bias or model-error estimation schemes using multiple, and sometimes collaborative sensors. In the radar signal processing literature, the process of model-based estimation and subsequent removal of systematic errors prior to target tracking is known as sensor-registration @cite_14 , @cite_9 . In the weather research literature, the serially-correlated forecasting error arising due to modeling deficiency is often considered separately and tracked alongside the model parameters @cite_32 , @cite_23 . In the sensor network literature, drift-aware networks perform learning-based collaborative bias estimation to enhance the effective lifetime of the network @cite_15 , @cite_12 . However, in this paper, we are focused on the quality of estimation in the presence of systematic errors, rather than techniques on mitigating systematic errors.
@cite_14 @cite_9 @cite_32 @cite_23 @cite_15 @cite_12
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Inaccuracies in the observation model of the synthetic aperture radar (SAR) due to inaccuracies of the velocity and position of the platform or atmospheric turbulence cause degradations in reconstructed images which necessitate the use of autofocus algorithms. In this paper we propose a novel signal processing algorithm for joint SAR image formation and autofocus in a synthesis dictionary based sparse representation framework. Proposed algorithm can be applied broadly to scenes that exhibit sparsity with respect to any dictionary. This is done by extending our previously developed sparse representation-based SAR imaging framework to joint SAR image formation and autofocus. To this end, the phase error vector is separated from the unknown phase of the complex-valued back-scattered field. Phase error vector is estimated using a MAP estimator and compensated through an iterative algorithm to produce focused images. We demonstrate the effectiveness of the proposed approach on synthetic and real imagery. An exact solution is provided for the multiple sensor bias estimation problem based on local tracks. It is shown that the sensor bias estimates can be obtained dynamically using the outputs of the local (biased) state estimators. This is accomplished by manipulating the local state estimates such that they yield pseudomeasurements of the sensor biases with additive noises that are zero-mean, white, and with easily calculated covariances. These results allow evaluation of the Cramer-Rao lower bound (CRLB) on the covariance of the sensor bias estimates, i.e., a quantification of the available information about the sensor biases in any scenario. Monte Carlo simulations show that this method has significant improvement in performance with reduced rms errors of 70 compared with commonly used decoupled Kalman filter. Furthermore, the new method is shown to be statistically efficient, i.e., it meets the CRLB. The extension of the new technique for dynamically varying sensor biases is also presented. Computational sensing strategies often suffer from calibration errors in the physical implementation of their ideal sensing models. Such uncertainties are typically addressed by using multiple, accurately chosen training signals to recover the missing information on the sensing model, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not employ any training signal, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models, in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors, each subject to an unknown positive multiplicative factor (or gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable, carefully chosen initialisation point. An analysis of this algorithm allows us to show that it converges to the exact solution provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Interestingly, we show that this requirement grows linearly (up to log factors) in the number of unknowns of the problem. This sample complexity is found both in absence of prior information, as well as when subspace priors are available for both the signal and gains, allowing a further reduction of the number of observations required for our recovery guarantees to hold. Moreover, in the presence of noise we show how our descent algorithm yields a solution whose accuracy degrades gracefully with the amount of noise affecting the measurements. Finally, we present some numerical experiments in an imaging context, where our algorithm allows for a simple solution to blind calibration of the gains in a sensor array. Wireless sensor networks are deployed for the purpose of monitoring an area of interest. Even when the sensors are properly calibrated at the time of deployment, they develop drift in their readings leading to erroneous network inferences. Based on the assumption that neighbouring sensors have correlated measurements and that the instantiations of drifts in sensors are uncorrelated, the authors present a novel algorithm for detecting and correcting sensor measurement errors. The authors use statistical modelling rather than physical relations to model the spatio-temporal cross-correlations among sensors. This in principle makes the framework presented applicable to most sensing problems. Each sensor in the network trains a support vector regression algorithm on its neighbours' corrected readings to obtain a predicted value for its future measurements. This phase is referred to here as the training phase. In the running phase, the predicted measurements are used by each node, in a recursive decentralised fashion, to self-assess its measurement and to detect and correct its drift and random error using an unscented Kalman filter. No assumptions regarding the linearity of drift or the density (closeness) of sensor deployment are made. The authors also demonstrate using real data obtained from the Intel Berkeley Research Laboratory that the proposed algorithm successfully suppresses drifts developed in sensors and thereby prolongs the effective lifetime of the network.