Signal approximation using the bilinear transform

Abstract

This thesis explores the approximation properties of a unique basis expansion. The expansion implements a nonlinear frequency warping between a continuous-time signal and its discrete-time representation according to the bilinear transform. Since there is a one-to-one mapping between the continuous-time and discrete-time frequency axes, the bilinear representation avoids any frequency aliasing distortions. We devote the first portion of this thesis to some theoretical properties of the bilinear representation, including the analysis and synthesis networks as well as bounds on the basis functions. These properties are crucial when we further analyze the bilinear approximation performance. We also consider a modified version of the bilinear representation in which the continuous-time signal is segmented using a short-duration window. This segmentation procedure affords greater time resolution and, in certain cases, improves the overall approximation quality. In the second portion of this thesis, we evaluate the approximation performance of the bilinear representation in two different applications. The first is approximating instrumental music. We compare the bilinear representation to a discrete cosine transform based approximation technique. The second application is computing the inner product of two continuous-time signals for a binary detection problem. In this case, we compare the bilinear representation with Nyquist sampling.