Moments and Wavelets in Signal Estimation
Edward J. Wegman, Hung T. Le, Wendy L. Poston, Jeffrey L. Solka · 2020
This chapter describes the basic wavelet theory in the context of the general statistical problem of nonparametric function estimation. It shows that traditional moment-based techniques have an interesting and useful connection to modern nonparametric functional inference for signal processing via wavelets. The chapter discusses the notion of a wavelet basis and demonstrates the connection with Fourier series and Parseval’s Theorem. The new methods of wavelet analysis have recently burst upon the mathematical scene to capture the enthusiasm and imagination of many applied mathematicians and engineers because of their important applications in signal processing, image analysis, pattern recognition, nonparametric function estimation, and other engineering applications and also because of the inherent elegance of the techniques. Probability density estimation and nonparametric, nonlinear regression are probably the two most widely studied nonparametric function estimation problems. A transient signal is a signal of finite duration, typically with a relatively sudden onset.