Determining distributed source waveforms in casual, lossy, dispersive, plane-wave (CLDP) materials

Abstract

This thesis presents and employs novel mathematics for the inversion of linear, first-kind Fredholm integral equations (IEs) which have a time t dependent response signal, a space z dependent source waveform, and a kernel with time dependence (at each z) corresponding to the impulse response of a thickness z slab of causal, lossy, dispersive, homogeneous material through which planar disturbances propagate according to the wave equation. These materials are called CLDP materials; these IEs are called CLDP IEs. These novel mathematics are applicable to the PESAW (aka PEA) charge recovery method. The proposed inversion method recognizes that the (temporal) Fourier transform of a CLDP IE's response signal can be interpreted as the values of the (spatial) Laplace transform of that IE's source waveform along a Laplace plane path determined by the material's propagation wavenumber k(f). Executing the Laplace transform inversion integral along this CLDP path yields an inverse CLDP IE which recovers the true source waveform provided that source waveform is real, causal, Fourier-transformable, and also satisfies the proposed k(f)-dependent 'CLDP criterion'. The forward and inverse CLDP IEs corresponding to a particular CLDP material model k(f) therefore comprise a particular integral transform relationship applicable to waveforms satisfying the CLDP criterion for that material. The CLDP transform relationship for a lossless/dispersionless material reduces to the (unilateral) Fourier transform. Even without noise, the 'inverse CLDP'-recovered waveform gleaned from an abruptly bandlimited CLDP response signal requires regularization - a generalized Gibbs-Dirichlet kernel dubbed 'the Darrell' comes into effect. The measured (time sampled) PESAW signal is necessarily bandlimited; this thesis investigates regularization via lowpass filtering of the measured signal. Both synthetic and experimental examples are investigated. The focus is on MHz-range signals culled from mm-range polymeric PESAW experiments. A method for determining the requisite model k(f) from measured PESAW signals is also presented and employed.