by Tam Yuk-ho. / Bibliography: leaves 93-94 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1986

Thesis (Ph.D.)--University of Delaware, 2007. / Principal faculty advisor: Kenneth E. Barner, Dept. of Electrical and Computer Engineering. Includes bibliographical references.

Graduation date: 1991

This report describes the design and implementation of a fixed audio equalizer based on a scheme where parts of the signal spectrum are downsampled and treated differently for the purpose of reducing the computational complexity and memory requirements. The primary focus has been on finding a way of taking an equalizer based on a simple minimum-phase FIR filter and transform it to the new type of equalizer. To achieve this, a number of undesireable effects such as aliasing distortion and upsampling imaging had to be considered and dealt with. In order to achieve a good amplitude response of the system, optimization procedures were used. As part of the thesis, a cost-effective implementation of the filter has been made for an FPGA, in order to verify that the scheme is indeed usable for equalizing an audio signal.

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In this study, we address the factorization problem in the Hardy H('p) spaces, and provide a fast algorithm for its implementation with applications to some important engineering problems. The Thesis is presented in three autonomous papers. / In the first paper we lay down the technical foundation of the new approach in the scalar case. First, the factorization problem is formulated in the H('p) spaces. A formulation with sufficient generality to encompass practically all such engineering problems. Necessary and sufficient conditions for the existence of the spectral factors are derived, and a characterization of the class of functions admitting a canonical factorization is obtained. The reduction method is applied to certain Toeplitz equations in H('2) space to generate a sequence of approximate spectral factors. When the Laguerre basis is used in the reduction method the Toeplitz equation turns out to a Toeplitz set of linear equations. We also provide an error bound and an estimate for the speed of convergence. / In the second paper the matrix version of all the scalar results is provided and enriched with discussions and extension. In particular, we have shown that the factorization problem is associated with the solutions of certain Toeplitz equations in / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / spaces. The classical Gohberg-Krein factorization is re-examined within the framework developed here, and the connections between the outer-factorization, the canonical factorization, and inversion of certain Toeplitz operators have also been unveiled. / In Part III we generalize the Davis and Barry formula for the feedback gain in the LQR problems. The new setting, equipped with the spectral factorization method, provides fast and efficient algorithms for solving a wide class of LQR problems, rational matrix factorization, and positive polynomials factorization. Our parallel results for the discrete time case are given in brief together with many interesting computational properties.

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