#### Date of Award

6-16-1998

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Department of Mathematics and Statistics

#### First Advisor

Gregory T. Warhola, PhD

#### Abstract

Representing speech signals such that specific characteristics of speech are included is essential in many Air Force and DoD signal processing applications. A mathematical construct called a frame is presented which captures the important time-varying characteristic of speech. Roughly speaking, frames generalize the idea of an orthogonal basis in a Hilbert space, Specific spaces applicable to speech are L_{2}(**R**) and the Hardy spaces H_{p}(**D**) for p> 1 where D is the unit disk in the complex plane. Results are given for representations in the Hardy spaces involving Carleson's inequalities (and its extensions), frames and hybrid frames, as well as L_{2}(**R**). Examples of different speech signals are given and the representations via frames are applied to demonstrate its robustness and adaptiveness, while using very few coefficients in the approximation. Thus, the processing, transmitting and storing of speech data could be compressed or reduced and still keep the fidelity of the signal.

#### AFIT Designator

AFIT-DS-ENC-98J-1

#### DTIC Accession Number

ADA348548

#### Recommended Citation

Suzuki, Laura R., "Representations, Approximations, and Algorithms for Mathematical Speech Processing" (1998). *Theses and Dissertations*. 5516.

https://scholar.afit.edu/etd/5516