Performance and Robustness Analysis of Co-Prime and Nested Sampling

Coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than the Nyquist rate, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This thesis studies the performance of coprime and nested samplers in spatial and temporal domains, when these assumptions are violated. In spatial domain, the robustness of these samplers is studied by considering arrays with perturbed sensor locations (with unknown perturbations). Simplified expressions for the Fisher Information matrix for perturbed coprime and nested arrays are derived, which explicitly highlight the role of co-array. It is shown that even in presence of perturbations, it is possible to resolve $O(M^2)$ under appropriate conditions on the size of the grid. The assumption of small perturbations leads to a novel ``bi-affine" model in terms of source powers and perturbations. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers. This thesis also studies the robustness of coprime sampling to finite number of samples and sampling jitter, by analyzing their effects on the quality of the estimated autocorrelation sequence. A variety of bounds on the error introduced by such non ideal sampling schemes are computed by considering a statistical model for the perturbation. They indicate that coprime sampling leads to stable estimation of the autocorrelation sequence, in presence of small perturbations. Under appropriate assumptions on the distribution of WSS signals, sharp bounds on the estimation error are established which indicate that the error decays exponentially with the number of samples. The theoretical claims are supported by extensive numerical experiments.

Diversity in Catalytic Reactions of Propargylic Diazoesters

Abstract Title of Document: Diversity in Catalytic Reactions of Propargylic Diazoesters Huang Qiu, Doctor of Philosophy, 2016 Directed By: Professor Michael P. Doyle, Department of Chemistry and Biochemistry Propargylic aryldiazoesters, which possess multiple reactive functional groups in a single molecule, were expected to undergo divergent reaction pathways as a function of catalysts. A variety of transition metal complexes including rhodium(II), palladium(II), silver(I), mercury(II), copper(I and II), and cationic gold (I) complexes have been examined to be effective in the catalytic domino reactions of propargylic aryldiazoesters. An unexpected Lewis acid catalyzed pathway was also discovered by using FeCl3 as the catalyst. Under the catalysis of selected gold catalysts, propargylic aryldiazoesters exist in equilibrium with 1-aryl-1,2-dien-1-yl diazoacetate allenes that are rapidly formed at room temperature through 1,3-acyloxy migration. The newly formed allenes further undergo a metal-free rearrangement in which the terminal nitrogen of the diazo functional group adds to the central carbon of the allene initiating a sequence of bond forming reactions resulting in the production of 1,5-dihydro-4H-pyrazol-4-ones in good yields. These 1,5-dihydro-4H-pyrazol-4-ones undergo intramolecular 1,3-acyl migration to form an equilibrium mixture or quantitatively transfer the acyl group to an external nucleophile with formation of 4-hydroxypyrazoles. In the presence of a pyridine-N-oxide, both E- and Z-1,3-dienyl aryldiazoacetates are formed in high combined yields by Au(I)-catalyzed rearrangement of propargyl arylyldiazoacetates at short reaction times. Under thermal reactions the E-isomers form the products from intramolecular [4+2]-cycloaddition with H‡298 = 15.6 kcal/mol and S‡298= -27.3 cal/ (mol•degree). The Z-isomer is inert to [4+2]-cycloaddition under these conditions. The Hammett relationships from aryl-substituted diazo esters ( = +0.89) and aryl-substituted dienes ( = -1.65) are consistent with the dipolar nature of this transformation. An unexpected reaction for the synthesis of seven-membered conjugated 1,4-diketones from propargylic diazoesters with unsaturated imines was disclosed. To undergo this process vinyl gold carbene intermediates generated by 1,2-acyloxy migration of propargylic aryldiazoesters undergo a formal [4+3]-cycloaddition, and the resulting aryldiazoesters tethered dihydroazepines undergo an intricate metal-free process to form observed seven-membered conjugated 1,4-diketones with moderate to high yields.