Applications of operator theory to time-frequency analysis and classification

A single discrete signal is associated with an essentially infinite set of quadratic time-frequency representations (TFRs) through appropriate choices for the kernel function. Useful kernels can be selected by incorporating into the kernel design properties that are desired in the end representation.; We present a kernel design procedure in which signal discrimination (in the time-frequency plane) is the only goal. Using our operator theory formulation for TFRs, we are able to easily develop a closed-form solution for an optimally discriminating kernel which is not signal dependent, but signal class dependent. Use of TFRs for signal classification and detection (a similar problem) has been researched previously, but from the point of view of discovering which, if any, of the existing TFRs might succeed with certain signal types. The idea of custom designing kernels has been explored, but not with an eye to classification.; We begin with a discussion of operator theory and our application of this to time-frequency analysis. We then discuss our kernel design procedure from both the time-frequency perspective and the ambiguity plane perspective which provides valuable insight into the nature of our operator theory formulation as well as our kernel design process. Examples of use are included.; With appropriate modifications and extensions, these representations could be suitable for use in a number of applications, among them machine tool monitoring. Included in this dissertation is an extensive discussion of our existing system for real-time monitoring of acoustic emissions ln machining applications. The goal of this work is to develop an approach to in-process monitoring which allows continuous assessment of tool wear and early warning of process exceptions. The nature of metal removal processes creates short-lived vibrations that carry information about the condition of the cutting tool and quality of cut. We wish to extract and represent these transient events without loss of important spectral structure. Other challenges include the need for system training data selection in the absence of expert labeled data, the modeling of short-term time evolution, and efficient real-time operation on an inexpensive computing platform. We present a system that meets these challenges through the use of high resolution time-frequency representations, vector quantization, hidden Markov models and a method of regrouping training data to refine initial class guesses. Applying this system to the classification of milling transients, we show that our system is capable of extracting these events and assigning them to meaningful classes--a crucial step in monitoring tool wear.