Two-Dimensional Signal Root Estimation in Blind Multichannel Impact Response Analysis

This doctoral dissertation concerns two-dimensional frequency estimation for modal analysis of single-input multiple-output impact responses from circularly cylindrical objects. The application is within detection of internal defects in trunks of spruce. The material consists of impact responses from 94 trees, with 66 sound and 28 in various stages of decay.

Paper I proposes that modal analysis of trunks of spruce may unveil the presence of internal defects, which are hypothesised to result in a reduction of the modal frequencies. The responses, collected using a uniform linear array of accelerometers encircling a cross-section, contain two-dimensional spatiotemporal modes which are sinusoidal and temporally damped. The spatial mode-shapes are used for ensuring that comparisons of modal frequencies on a tree-to-tree basis are performed on corresponding modes. The frequency analysis is based on cascading of a polynomial-based estimator for the temporal dimension and DFTs for the spatial dimension. The results show that the modal frequencies often are lower for trees in decay than for sound trees, but also, that the groups partially overlap, and that the results for sound trees with matching dimensions are scattered.

Paper II is based on the results in Paper I, and addresses a method for reduction of scatter within the group of sound trees, using one additional parameter that can be estimated from the impact responses. Furthermore, a detector is proposed, and its performance is found to be better than performance reported elsewhere for visual tree examination.

Paper III presents a novel technique, called partial forward-backward averaging, that enhances two-dimensional state-space/subspace rotation-based frequency estimation for the kind of modes encountered in the current application. Conventional estimators do not account for all symmetries inherent in this particular kind of modes, and consequently, the quality of the resulting estimates is poor. The novel technique, model-based enhancement of signal subspace estimation, results in estimates with better properties and lower bias and variance.

Paper IV contains a full re-analysis of, and a comparison with, the material reported on in Paper I. A fast implementation of a state-space/subspace rotation-based algorithm, detailed in Paper III, is derived and used for the analysis. It has been hypothesised that the modal frequencies change when a tree is subjected to impact excitation repeatedly, since the impacts are applied onto a screw which inevitably is forced deeper and deeper into the trunk. No such effect is found, so this hypothesis is rejected.