#### Title

#### DOI

10.17077/etd.d0za6wct

#### Document Type

Dissertation

#### Date of Degree

Fall 2014

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Electrical and Computer Engineering

#### First Advisor

Soura Dasgupta

#### Second Advisor

Weiyu Xu

#### First Committee Member

Soura Dasgupta

#### Second Committee Member

Weiyu Xu

#### Third Committee Member

Raghuraman Mudumbai

#### Fourth Committee Member

Erwei Bai

#### Fifth Committee Member

Paul Muhly

#### Abstract

Location information is of extreme importance in every walk of life ranging from commercial applications such as location based advertising and location aware next generation communication networks such as the 5G networks to security based applications like threat localization and E-911 calling. In indoor and dense urban environments plagued by multipath effects there is usually a Non Line of Sight (NLOS) scenario preventing GPS based localization. Wireless localization using sensor networks provides a cost effective and accurate solution to the wireless source localization problem. Certain sensor geometries show significantly poor performance even in low noise scenarios when triangulation based localization methods are used. This brings the need for the design of an optimum sensor placement scheme for better performance in the source localization process.

The optimum sensor placement is the one that optimizes the underlying Fisher Information Matrix(FIM) . This thesis will present a class of canonical optimum sensor placements that produce the optimum FIM for N-dimensional source localization N greater than or equal to 2 for a case where the source location has a radially symmetric probability density function within a N-dimensional sphere and the sensors are all on or outside the surface of a concentric outer N-dimensional sphere. While the canonical solution that we designed for the 2D problem represents optimum spherical codes, the study of 3 or higher dimensional design provides great insights into the design of measurement matrices with equal norm columns that have the smallest possible condition number. Such matrices are of importance in compressed sensing based applications.

This thesis also presents an optimum sensing matrix design for energy efficient source localization in 2D. Specifically, the results relate to the worst case scenario when the minimum number of sensors are active in the sensor network. We also propose a distributed control law that guides the motion of the sensors on the circumference of the outer circle so that achieve the optimum sensor placement with minimum communication overhead.

The design of equal norm column sensing matrices has a variety of other applications apart from the optimum sensor placement for N-dimensional source localization. One such application is fourier analysis in Magnetic Resonance Imaging (MRI). Depending on the method used to acquire the MR image, one can choose an appropriate transform domain that transforms the MR image into a sparse image that is compressible. Some such transform domains include Wavelet Transform and Fourier Transform. The inherent sparsity of the MR images in an appropriately chosen transform domain, motivates one of the objectives of this thesis which is to provide a method for designing a compressive sensing measurement matrix by choosing a subset of rows from the Discrete Fourier Transform (DFT) matrix. This thesis uses the spark of the matrix as the design criterion. The spark of a matrix is defined as the smallest number of linearly dependent columns of the matrix. The objective is to select a subset of rows from the DFT matrix in order to achieve maximum spark. The design procedure leads us to an interest study of coprime conditions on the row indices chosen with the size of the DFT matrix.

#### Public Abstract

Location information is of extreme importance in every walk of life ranging from commercial applications such as location based advertising and location aware next generation communication networks such as the 5G networks to security based applications like threat localization and E-911 calling. Locating a source of threat or a missing person using wireless signals transmitted from the source are examples of daily life situations that critically rely on location information. In indoor and dense urban environments the wireless signals are reflected/scattered by the obstacles such as buildings and trees. This degrades the quality of the wireless signal from the source that we are looking for. This negatively impacts the accuracy of the localization process. Thus one needs to understand the wireless environment and smartly deploy the sensors at appropriate places in order to get an accurate location estimate. In this thesis, the design of the optimum sensor deployment is modeled as the design of matrices with the columns defined as the sensor locations such that these matrices have certain optimum matrix properties.

Another interesting application where these matrix design techniques can be used is Magnetic Resonance Imaging. The motivation to study matrix design techniques for MRI comes from the fact that we would like to make few acquisitions but yet be able to understand the missing portions of the data. We would like to design a matrix which has optimum properties such that the measurement process of the MRI modeled by this matrix retains all the important information required.

#### Keywords

publicabstract, Compressed Sensing, Fisher Information Matrix, Localization, Optimum Sensor Placement, Sensing Matrices

#### Pages

xiii, 148 pages

#### Bibliography

Includes bibliographical references (pages 143-148).

#### Copyright

Copyright 2014 Hema Kumari Achanta

#### Recommended Citation

Achanta, Hema Kumari. "Optimal sensing matrices." PhD (Doctor of Philosophy) thesis, University of Iowa, 2014.

https://doi.org/10.17077/etd.d0za6wct.