This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard matrices) and wavelet reconstruction the stable sampling rate is linear. This implies that binary measurements are as efficient as Fourier samples when using wavelets as the reconstruction space. Powerful techniques for reconstructions include generalized sampling and its compressed versions, as well as recent methods based on data assimilation. Common to these methods is that the reconstruction quality depends highly on the subspace angle between the sampling and the reconstruction space, which is dictated by the stable sampling rate. As a result of the theory provided in this paper, these methods can now easily use binary measurements and wavelet reconstruction bases.

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Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has long been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Binary measurements are a mainstay in signal and image processing and can be modelled by the Walsh transform and Walsh series that are binary cousins of the respective Fourier counterparts. We help bridging the theoretical gap by providing non-uniform recovery guarantees for infinite-dimensional compressed sensing with Walsh samples and wavelet reconstruction. The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case. However, there is a fundamental difference in the asymptotic results when the smoothness and vanishing moments of the wavelet increase. In the Fourier case, this changes the optimal sampling patterns, whereas this is not the case in the Walsh setting.

This research was conducted within the framework of a National Science Foundation sponsored summer Research Experience for Undergraduate (REU) students. This research considers small-scale and mathematical models of simple one-story structures that are subjected to free and base-motion excitations and installed with and without passive damping devices to gain an understanding of their dynamic behavior while reviewing active and semi-active damping means being applied and researched today. Using computer programming and numerical methods, the goal is to understand and counteract catastrophic disasters to structures caused by earthquakes. The research is broken down into a number of MATLAB simulations and experiments in order to understand basic dynamic and control features required to design earthquake resilient buildings. These experiments include free vibration experiments to test for the stiffness of columns for different heights and to test for the natural frequency and damping ratio of a one-story structure under different mass loads. Active PD control was then applied to an experimental system experiencing accelerations attributed to the Northridge 1994, Kobe 1995, El Centro 1940, and Mendocino 1992 earthquakes. Robustness comparisons were made between (1) P control; (2) D control; and (3) PD control for the above earthquake inputs to the shaker. A fuzzy logic controller was developed to effectively control transient vibrations. The uniqueness of this control concept is that the fuzzy control continuously varies the damping characteristics of a semi-active tuned mass damper (TMD). It was concluded that a fuzzy logic based TMD was more effective than a regular passive TMD, by providing half the settling times. ß

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