Analytical derivation of lower bounds of the estimation error of statistically unbiased estimation of electromagnetic/acoustic wireless signal parameters

Abstract

This dissertation presents four related investigations: 1. Direction finding using an array of isotropic sensors with gains of Bayesian uncertainty -the hybrid Cram´er-Rao bound and maximum a posteriori estimation. This investigation, discussed in Chapter 3, analyzes how a sensor array's direction-finding accuracy may be degraded by any stochastic uncertainty in the sensors' gains. The analysis is through derivation of hybrid Cram´er-Rao bound of azimuth-elevation direction-of-arrival estimates. This work is first in the open literature to derive the hybrid Cram´er-Rao bound of azimuth-elevation direction-of-arrival estimates of a signal incident upon an array of isotropic sensors that suffer gain uncertainties. This hybrid Cram´er-Rao bound is analytically shown to be inversely proportional to a multiplicative factor equal to one plus the variance of the sensors' gain uncertainty. This finding applies to any array-grid geometry. Monte Carlo simulations demonstrate that the performance of maximum a posteriori estimator approaches the lower bound derived. This work has been published in IEEE Transactions of Aerospace and Electronic Systems (authors -the candidate, his chief supervisor and two other collaborators). 2. Hybrid Cram´er-Rao bound for direction-of-arrival estimation, at a triad of higher-order cardioid sensors in perpendicular orientation and spatial colocation. This investigation, discussed in Chapter 4, pioneers the idea of placing three cardioid microphones/hydrophones, of any cardioidic order, in orthogonal orientation but spatial colocation. The triad's azimuth-elevation direction-of-arrival estimation accuracy is studied. This accuracy is measured via hybrid Cram´er-Rao bound. The cardiodicity index is allowed to be stochastically uncertain. Given in closed form, and comprehensively analyzed, is the hybrid Cram´er-Rao bound corresponding to a first-order cardioid. Monte Carlo simulations show that the performance of maximum a posteriori estimator approaches the lower bound derived. This work is under review by the Journal of Sound and Vibration (authors -the candidate, his chief supervisor and two other collaborators). 3. Hybrid Cram´er-Rao bound for estimation of direction-of-arrival and polarization, using an electromagnetic dipole/loop antenna triad with Bayesian mutual coupling. This investigation, discussed in Chapter 5, introduces the concept of mutual coupling across a triad of electromagnetic dipole/loop antennas, in direction-of-arrival and polarization parameters' estimation. A variety of signal processing algorithms have previously been designed to study polarization and direction-of-arrival of electromagnetic waves, using dipole/loop antennas. However, none of the previous investigations has taken into account the fact that each of the constituent dipole/loop antenna may experience electromagnetic coupling among the other antennas. There is thus a need to investigate this practical factor of mutual coupling among the dipole/loop antennas. This investigation/analysis is studied here through derivation of the hybrid Cram´er-Rao bound for the azimuth-elevation direction-of-arrival and the polarization parameters, using an electromagnetic dipole/loop antenna triad with mutual impedance among the antennas. The mutual impedance is considered to be stochastically uncertain, with a priori known mean and variance. 4. Cram´er-Rao bound for direction-of-arrival estimation at a tri-axial velocity sensor with inter-channel incoherence. This investigation, discussed in Chapter 6, considers statistical incoherence of data across the three channels of a tri-axial velocity sensor. All previous direction-of-arrival estimation algorithms for a tri-axial velocity sensor presume the tri-axial velocity sensor's data to be statistically coherent across the three channels. However, the channels can become de-correlated. This investigation comes in to fill such missing gap. The analysis on how the incoherence of the received data (or rather de-correlation of the channels) affects the direction-of-arrival estimation accuracy, is done through derivation of Cram´er-Rao bound for the azimuth-elevation direction-of-arrival. Monte Carlo simulations show that the performance of maximum likelihood estimator approaches the derived lower bounds.