Representations, Approximations, and Algorithms for Mathematical Speech Processing

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\u3e 1 where D is the unit disk in the complex plane. Results are given for representations in the Hardy spaces involving Carleson\u27s 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