Spectral theory (Mathematics)

In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical
theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.

On this thesis, we present a detailed study of
the exact sequence of terms of low degree, which obtains
when the spectral systems being considered are
suitably restricted. Its validity is established
under more general conditions than could be found in
the literature on spectral sequences.

We characterize the Multitaper Spectral Estimation method as a tool for stationary signal analysis. We compare its performance to the conventional periodogram, the parametric autoregressive and multitaper autoregressive spectral estimates. We analyze single and two frequency sinusoids with additive Gaussian white noise and autoregressive processes of orders 2, 4 and 24. We extend its application to non-stationary signals and develop the multitaper spectrogram. We test the spectrograms with simulated non-stationary autoregressive process of order 2 as the magnitude of its poles vary between 0 and 1 and the angle of the poles vary between 0 and pi. Our results show that the multitaper spectral estimate can be parameterized and is more accurate than others tested for non-sinusoidal signals. We also show applications to aero-acoustic data analysis.

In this thesis, a 2D CHebyshev spectral domain decomposition method is developed for simulating the generation and propagation of internal waves over a topography. While the problem of stratified flow over topography is by no means a new one, many aspects of internal wave generation and breaking are still poorly understood. This thesis aims to reproduce certain observed features of internal waves by using a Chebyshev collation method in both spatial directions. The numerical model solves the inviscid, incomprehensible, fully non-linear, non-hydrostatic Boussinesq equations in the vorticity-streamfunction formulation. A number of important features of internal waves over topography are captured with the present model, including the onset of wave-breaking at sub-critical Froude numbers, up to the point of overturning of the pycnoclines. Density contours and wave spectra are presented for different combinations of Froude numbers, stratifications and topographic slope.

The goal of a speech enhancement algorithm is to remove noise and recover the original signal with as little distortion and residual noise as possible. Most successful real-time algorithms thereof have done in the frequency domain where the frequency amplitude of clean speech is estimated per short-time frame of the noisy signal. The state of-the-art short-time spectral amplitude estimator algorithms estimate the clean spectral amplitude in terms of the power spectral density (PSD) function of the noisy signal. The PSD has to be computed from a large ensemble of signal realizations. However, in practice, it may only be estimated from a finite-length sample of a single realization of the signal. Estimation errors introduced by these limitations deviate the solution from the optimal. Various spectral estimation techniques, many with added spectral smoothing, have been investigated for decades to reduce the estimation errors. These algorithms do not address significantly issue on quality of speech as perceived by a human. This dissertation presents analysis and techniques that offer spectral refinements toward speech enhancement. We present an analytical framework of the effect of spectral estimate variance on the performance of speech enhancement. We use the variance quality factor (VQF) as a quantitative measure of estimated spectra. We show that reducing the spectral estimator VQF reduces significantly the VQF of the enhanced speech. The Autoregressive Multitaper (ARMT) spectral estimate is proposed as a low VQF spectral estimator for use in speech enhancement algorithms. An innovative method of incorporating a speech production model using multiband excitation is also presented as a technique to emphasize the harmonic components of the glottal speech input.