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.
Several researchers have studied the Cram 'er-Rao bounds for polynomial (or polyphase) signal estimation in the presence of independent (or correlated) noise. The CRB is usually obtained from the inverse of the Fisher Information Matrix (FIM) @cite_33 . For Additive White Gaussian Noise (AWGN), the large sample approximation of FIM is known to be a multiple of the Hilbert Matrix (e.g., @cite_27 ). The second order approximation was derived in @cite_19 in the context of polynomial phase signals. For a mixture of additive and multiplicative white Gaussian noise, the large sample FIM was shown in @cite_2 to be a scalar multiple of the AWGN case. . We also discuss the non-stationary case when the autoregressive parameter is equal to @math . To the knowledge of the authors, polynomial signal estimation in such a mixture of noise has not been considered earlier. As mentioned earlier, the study of estimating polynomial signals in AR(1)+White noise would help us characterize the capability of a sensor network to infer the parameters of a time-varying signal using sensors with drift.
@cite_19 @cite_27 @cite_33 @cite_2
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The authors derive the Cramer-Rao lower bound (CRLB) for complex signals with constant amplitude and polynomial phase, measured in additive Gaussian white noise. The exact bound requires numerical inversion of an ill-conditioned matrix, while its O(N sup -1 ) approximation is free of matrix inversion. The approximation is tested for several typical parameter values and is found to be excellent in most cases. The formulas derived are of practical value in several radar applications, such as electronic intelligence systems (ELINT) for special pulse-compression radars, and motion estimation from Doppler measurements. Consequently, it is of interest to analyze the best possible performance of potential estimators of the phase coefficients, as a function of signal parameters, the signal-to-noise ratio, the sampling rate, and the number of measurements. This analysis is carried out. > Abstract We derive expressions for the Cramer-Rao Bound (CRB) for the estimates of the parameters of a deterministic signal observed in additive and multiplicative noise which may be i.i.d. non-Gaussian or colored Gaussian. General expressions for the CRB are derived, and are then applied to the specific case of polynomial phase signals for which closed-form expressions are obtained under certain assumptions. We also develop bounds on the CRB; these bounds are tight at low SNR. 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. This paper studies the estimation of a high-dimensional vector signal where the observation is a known "sparse" linear transformation of the signal corrupted by additive Gaussian noise. A paradigm of such a linear system is code-division multiple access (CDMA) channel with sparse spreading matrix. Assuming a "semi-regular" ensemble of sparse matrix linear transformations, where the bi-partite graph describing the system is asymptotically cycle-free, it is shown that belief propagation (BP) achieves the minimum mean-square error (MMSE) in estimating the transformation of the input vector in the large-system limit. The result holds regardless of the the distribution and power of the input symbols. Furthermore, the mean squared error of estimating each symbol of the input vector using BP is proved to be equal to the MMSE of estimating the same symbol through a scalar Gaussian channel with some degradation in the signal-to-noise ratio (SNR). The degradation, called the efficiency, is determined from a fixed-point equation due to Guo and Verdu, which is a generalization of Tanaka's formula to arbitrary prior distributions.