This research considers the Fourier transform calculations of multidimensional signals. The calculations are based on random sampling, where the sampling points are nonuniformly distributed according to strategically selected probability functions, to provide new opportunities that are unavailable in the uniform sampling environment. The latter imposes the sampling density of at least the Nyquist density. Otherwise, alias frequencies occur in the processed bandwidth which can lead to irresolvable processing problems. Random sampling can mitigate Nyquist limit that classical uniform-sampling-based approaches endure, for the purpose of performing direct (with no prefiltering or downconverting) Fourier analysis of (high-frequency) signals with unknown spectrum support using low sampling density. Lowering the sampling density while achieving the same signal processing objective could be an efficient, if not essential, way of exploiting the system resources in terms of power, hardware complexity and the acquisition-processing time.
In this research we investigate and devise novel random sampling estimation schemes for multidimensional Fourier transform. The main focus of the investigation and development is on the aspect of the quality of estimated Fourier transform in terms of the sampling density. The former aspect is crucial as it serves towards the heart objective of random sampling of lowering the sampling density. This research was motivated by the applicability of the random-sampling-based approaches in determining the Fourier transform in multidimensional Nuclear Magnetic Resonance (NMR) spectroscopy to resolve the critical issue of its long experimental time
