Full wave models in linear time invariant signal processing

Abstract

Full wave models have assumed a niche role in signal processing due to their complexity, limitation to (mostly) empty space, often absent sources, lack of intuition, and insufficient integration with the theory of signal processing in general. This thesis removes these shortcomings and derives closed form solutions for source generated wave fields in the presence of arbitrarily shaped objects. This is made possible by including absorption effects, substantially augmenting the eigenfunction expansion method, and combining both with linear system theory. The result is a generic closed form state space model and impulse response formula which reduces wave models to linear filters, both theoretically and practically, yet contains only elementary functions. The reasons for this is the explicit use of rectangular regions which also furnishes a considerably simpler derivation of some existing results like dimensionality scaling laws because the corresponding Fourier series naturally connects dimensionality with signal bandwidth. Furthermore, the thesis uses the Fourier series to link boundary value problems with information theory and shows how to store and retrieve information in a wave field, exemplified by an elastic rubber string. Despite simplifying some existing results, the main focus is on source generated wave fields and their connection to linear filters, as underlined with closed form solutions for waves in a wave guide, around the human head, in an urban street area, and more. These examples also emphasise how the results from this thesis allow everyone with a signal processing - but not necessarily a Partial Differential Equation (PDE) - background to use full wave models because these have been reduced to linear filters with a closed form impulse- and frequency response. This makes full wave models less intimidating to approach, adds intuition, and reduces many complex wave problems to simpler ones, as shown for the source localisation problem in highly reverberant environments. The thesis also extends the reach of signal processing beyond standard wave problems to include linearised water models, bio-heat transfer, chemical diffusion, and more in the same fashion.