UC San Diego

The objective of this thesis is to develop a few approaches to 'wave theory of information'. Specifically, this dissertation focuses on two special types of waveforms, bandlimited and multi-band signals. In both cases, we investigate the waveforms in the context of signal analysis and reconstructions.

In the first part of this thesis, we derive the amount of information that can be transmitted by bandlimited waveforms under perturbation, and the amount of information required to represent any bandlimited waveforms within a specific accuracy. These goals can be studied using a stochastic approach or a deterministic approach.Despite their shared goal of mathematically describing communication using the transmission of waveforms, as well as the common geometric intuition behind their arguments, the two approaches to information theory have evolved separately. The stochastic approach flourished in the context of communication, becoming the pillar of modern digital technologies, while the deterministic approach impacted mostly mathematical analysis. Recent interest in deterministic models has been raised in the context of networked control theory. This brings renewed attention to the deterministic approach in information theory. However, in contrast with the stochastic approaches where the tight results are already known, the previous deterministic results only provide the loose bounds. We improve these results by deriving tight results, and compare our results with the stochastic ones, which reveals the intrinsic similarities of two different approaches.

In the second part of this dissertation, we derive the minimum number of measurements to reconstruct multi-band waveforms, without any spectral information aside from the measure of the whole support set in the frequency domain. This problem is called the completely blind sensing problem and has been an open question. Until a recent date, partially blind sensing has been performed commonly instead, assuming to have some partial spectral information available a priori. We provide an answer for the completely blind sensing problem by deriving the minimum number of measurements to guarantee the reconstruction. The blind sensing problem shares some similarities with the compressed sensing problem. Despite these similarities, due to their different settings, the blind sensing problem contains a few additional difficulties which are not included in the compressed sensing problem. We independently develop our own theory to solve the completely blind sensing problem, and compare our results to those of the compressed sensing problem to reveal the similarities and differences between the two problems.