Matching Items (5)
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- Creators: Cochran, Douglas
Description
Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations.
ContributorsRobinson, Benjamin (Author) / Cochran, Douglas (Thesis advisor) / Moran, William (Thesis advisor) / Boggess, Albert (Committee member) / Milner, Fabio (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2014
Description
Modern measurement schemes for linear dynamical systems are typically designed so that different sensors can be scheduled to be used at each time step. To determine which sensors to use, various metrics have been suggested. One possible such metric is the observability of the system. Observability is a binary condition determining whether a finite number of measurements suffice to recover the initial state. However to employ observability for sensor scheduling, the binary definition needs to be expanded so that one can measure how observable a system is with a particular measurement scheme, i.e. one needs a metric of observability. Most methods utilizing an observability metric are about sensor selection and not for sensor scheduling. In this dissertation we present a new approach to utilize the observability for sensor scheduling by employing the condition number of the observability matrix as the metric and using column subset selection to create an algorithm to choose which sensors to use at each time step. To this end we use a rank revealing QR factorization algorithm to select sensors. Several numerical experiments are used to demonstrate the performance of the proposed scheme.
ContributorsIlkturk, Utku (Author) / Gelb, Anne (Thesis advisor) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Renaut, Rosemary (Committee member) / Armbruster, Dieter (Committee member) / Arizona State University (Publisher)
Created2015
Description
The solution of the linear system of equations $Ax\approx b$ arising from the discretization of an ill-posed integral equation with a square integrable kernel is considered. The solution by means of Tikhonov regularization in which $x$ is found to as the minimizer of $J(x)=\{ \|Ax -b\|_2^2 + \lambda^2 \|L x\|_2^2\}$ introduces the unknown regularization parameter $\lambda$ which trades off the fidelity of the solution data fit and its smoothing norm, which is determined by the choice of $L$. The Generalized Discrepancy Principle (GDP) and Unbiased Predictive Risk Estimator (UPRE) are methods for finding $\lambda$ given prior conditions on the noise in the measurements $b$. Here we consider the case of $L=I$, and hence use the relationship between the singular value expansion and the singular value decomposition for square integrable kernels to prove that the GDP and UPRE estimates yield a convergent sequence for $\lambda$ with increasing problem size. Hence the estimate of $\lambda$ for a large problem may be found by down-sampling to a smaller problem, or to a set of smaller problems, and applying these estimators more efficiently on the smaller problems. In consequence the large scale problem can be solved in a single step immediately with the parameter found from the down sampled problem(s).
ContributorsHorst, Michael Jacob (Author) / Renaut, Rosemary (Thesis director) / Cochran, Douglas (Committee member) / Wang, Yang (Committee member) / Barrett, The Honors College (Contributor) / School of Music (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-05
Description
The use of conventional weather radar in vulcanology leads to two problems: the radars often use wavelengths which are too long to detect the fine ash particles, and they cannot be field–adjusted to fit the wide variety of eruptions. Thus, to better study these geologic processes, a new radar must be developed that is easily reconfigurable to allow for flexibility and can operate at sufficiently short wavelengths.
This thesis investigates how to design a radar using a field–programmable gate array board to generate the radar signal, and process the returned signal to determine the distance and concentration of objects (in this case, ash). The purpose of using such a board lies in its reconfigurability—a design can (relatively easily) be adjusted, recompiled, and reuploaded to the hardware with none of the cost or time overhead required of a standard weather radar.
The design operates on the principle of frequency–modulated continuous–waves, in which the output signal frequency changes as a function of time. The difference in transmit and echo frequencies determines the distance of an object, while the magnitude of a particular difference frequency corresponds to concentration. Thus, by viewing a spectrum of frequency differences, one is able to see both the concentration and distances of ash from the radar.
The transmit signal data was created in MATLAB®, while the radar was designed with MATLAB® Simulink® using hardware IP blocks and implemented on the ROACH2 signal processing hardware, which utilizes a Xilinx® Virtex®–6 chip. The output is read from a computer linked to the hardware through Ethernet, using a Python™ script. Testing revealed minor flaws due to the usage of lower–grade components in the prototype. However, the functionality of the proposed radar design was proven, making this approach to radar a promising path for modern vulcanology.
This thesis investigates how to design a radar using a field–programmable gate array board to generate the radar signal, and process the returned signal to determine the distance and concentration of objects (in this case, ash). The purpose of using such a board lies in its reconfigurability—a design can (relatively easily) be adjusted, recompiled, and reuploaded to the hardware with none of the cost or time overhead required of a standard weather radar.
The design operates on the principle of frequency–modulated continuous–waves, in which the output signal frequency changes as a function of time. The difference in transmit and echo frequencies determines the distance of an object, while the magnitude of a particular difference frequency corresponds to concentration. Thus, by viewing a spectrum of frequency differences, one is able to see both the concentration and distances of ash from the radar.
The transmit signal data was created in MATLAB®, while the radar was designed with MATLAB® Simulink® using hardware IP blocks and implemented on the ROACH2 signal processing hardware, which utilizes a Xilinx® Virtex®–6 chip. The output is read from a computer linked to the hardware through Ethernet, using a Python™ script. Testing revealed minor flaws due to the usage of lower–grade components in the prototype. However, the functionality of the proposed radar design was proven, making this approach to radar a promising path for modern vulcanology.
ContributorsLee, Byeong Mok (Co-author) / Xi, Andrew Jinchi (Co-author) / Groppi, Christopher (Thesis director) / Mauskopf, Philip (Committee member) / Baumann, Alicia (Committee member) / Cochran, Douglas (Committee member) / Electrical Engineering Program (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
The use of conventional weather radar in vulcanology leads to two problems: the radars often use wavelengths which are too long to detect the fine ash particles, and they cannot be field–adjusted to fit the wide variety of eruptions. Thus, to better study these geologic processes, a new radar must be developed that is easily reconfigurable to allow for flexibility and can operate at sufficiently short wavelengths.
This thesis investigates how to design a radar using a field–programmable gate array board to generate the radar signal, and process the returned signal to determine the distance and concentration of objects (in this case, ash). The purpose of using such a board lies in its reconfigurability—a design can (relatively easily) be adjusted, recompiled, and reuploaded to the hardware with none of the cost or time overhead required of a standard weather radar.
The design operates on the principle of frequency–modulated continuous–waves, in which the output signal frequency changes as a function of time. The difference in transmit and echo frequencies determines the distance of an object, while the magnitude of a particular difference frequency corresponds to concentration. Thus, by viewing a spectrum of frequency differences, one is able to see both the concentration and distances of ash from the radar.
The transmit signal data was created in MATLAB®, while the radar was designed with MATLAB® Simulink® using hardware IP blocks and implemented on the ROACH2 signal processing hardware, which utilizes a Xilinx® Virtex®–6 chip. The output is read from a computer linked to the hardware through Ethernet, using a Python™ script. Testing revealed minor flaws due to the usage of lower–grade components in the prototype. However, the functionality of the proposed radar design was proven, making this approach to radar a promising path for modern vulcanology.
This thesis investigates how to design a radar using a field–programmable gate array board to generate the radar signal, and process the returned signal to determine the distance and concentration of objects (in this case, ash). The purpose of using such a board lies in its reconfigurability—a design can (relatively easily) be adjusted, recompiled, and reuploaded to the hardware with none of the cost or time overhead required of a standard weather radar.
The design operates on the principle of frequency–modulated continuous–waves, in which the output signal frequency changes as a function of time. The difference in transmit and echo frequencies determines the distance of an object, while the magnitude of a particular difference frequency corresponds to concentration. Thus, by viewing a spectrum of frequency differences, one is able to see both the concentration and distances of ash from the radar.
The transmit signal data was created in MATLAB®, while the radar was designed with MATLAB® Simulink® using hardware IP blocks and implemented on the ROACH2 signal processing hardware, which utilizes a Xilinx® Virtex®–6 chip. The output is read from a computer linked to the hardware through Ethernet, using a Python™ script. Testing revealed minor flaws due to the usage of lower–grade components in the prototype. However, the functionality of the proposed radar design was proven, making this approach to radar a promising path for modern vulcanology.
ContributorsXi, Andrew Jinchi (Co-author) / Lee, Matthew Byeongmok (Co-author) / Groppi, Christopher (Thesis director) / Mauskopf, Philip (Committee member) / Cochran, Douglas (Committee member) / Baumann, Alicia (Committee member) / Electrical Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05