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.

The Application of an Exogenous Linear and Radial Electrical Field to an In Vitro Chronic Diabetic Ulcer Model for Evaluation as a Potential Treatment

Chronic diabetic ulcers affect approximately 15% of patients with diabetes worldwide. Currently, applied electric fields are being investigated as a reliable and cost-effective treatment. This in vitro study aimed to determine the effects of a constant and spatially variable electric field on three factors: endothelial cell migration, proliferation, and angiogenic gene expression. Results for a constant electric field of 0.01 V demonstrated that migration at short time points increased 20-fold and proliferation at long time points increased by a factor of 1.40. Results for a spatially variable electric field did not increase directional migration, but increased proliferation by a factor of 1.39 and by a factor of 1.55 after application of 1.00 V and 0.01 V, respectively. Both constant and spatially variable applied fields increased angiogenic gene expression. Future research that explores a narrower range of intensity levels may more clearly identify the optimal design specifications of a spatially variable electric field.