## Optical Science and Engineering ETDs

9-12-2014

#### Abstract

In a communication system, a signal carrying information about a physical variable, or parameter, is fed into the front end of a channel and noisy or corrupted data are obtained at its back end, data from which one attempts to estimate the physical parameter. The error that accompanies such an estimate is usually characterized by mean squared error (MSE). In a Bayesian setting, the minimum value of the MSE is obtained by the so-called MMSE estimator (MMSEE), which in this sense is the best possible estimator. In general, the MMSE as well as the MMSEE are difficult to compute, which gave early impetus to work on a host of bounds on the MMSE of varying degrees of tightness over the last fifty years. The bounds, if sufficiently tight, help us in evaluating the performance of sub-optimal estimators. A widely used lower bound on the MMSE is the Ziv-Zakai lower bound, which bounds the MMSE via the minimum probability of error (MPE) of a binary hypothesis testing problem. Extensions of the Ziv-Zakai lower bound and some computationally efficient approximations are derived. The extensions include a bound in terms of the MPE of an M-ary hypothesis testing problem and another bound for discrete prior probability distributions. A major focus of the dissertation has been on deriving tight upper bounds on the MMSE, and their applications to imaging and non-imaging problems. An upper bound on the MMSE is derived that has a variational character, is easy to compute, and follows the MMSE tightly. This new upper bound on the MMSE is shown to be the bias removed mean squared error'' of {\\it any} test estimator. By choosing test estimators from suitably parameterized families and then optimizing over the parameters of the families, we obtain tight upper bounds on the MMSE as well as the estimators that achieve the upper bound. A new piecewise quasi-linear estimator which works well particularly when the data depend non-linearly on the parameters being estimated is proposed. The MMSE upper bound for such estimators performs tightly in bounding the MMSE for all values of the signal to noise ratio (SNR). The upper bounds obtained from the test estimators are applied to illustrative, toy 1-pixel and 2-pixel models of the EM-CCD camera. Further illustrations are provided by applying the upper bound to a highly non-linear and more realistic time-delay estimation (TDE) problem which is basic to many signal processing scenarios, including source detection, array processing, surveillance, synchronization in communications, range finding in RADAR, geo-location and tracking in sensor networks. A comparison with the existing bounds on the MMSE shows that our upper bound performs optimally in terms of its tightness to the MMSE and is the best of all bounds that we considered in all regions of operation. The upper bound was finally applied to characterize the performance of a newly developed rotating point spread function imaging system, which is capable of snapshot 3D imaging, and compared with the performance of a conventional imaging system for localizing point sources beyond the diffraction limit. New asymptotic analyses of the MMSE are also studied, both in the low SNR and the high SNR regimes, and a useful interpolational approximation of the MMSE is presented which seems to approximate accurately the MMSE for all values of the SNR.

#### Degree Name

Optical Science and Engineering

Doctoral

#### Department Name

Optical Science and Engineering

Hayat, Majeed

Lidke, Keith

Thomas, James

Dissertation

English

COinS