9 - Linear Equations, Inverse Filters, and Signal Analysis

Abstract

This chapter describes the linear equation models for inverse filters, signal analysis, linear and orthogonal regressions, and principal component analysis. Signal processing using a finite number of data records can be formulated by linear equations. The least square error criterion is quite an important basis for solving the linear equations that are employed in acoustic signal processing. The pseudo-inverse matrix using the singular value decomposition is important, particularly for finding the least square error solution. The chapter does not describe the elements of matrix theory. However, it encourages the study of the fundamentals of the mathematical theories of the linear equations for having a better understanding of acoustic signal processing. It describes inverse filtering and signal analysis based on the discrete models. These problems can be formulated using linear simultaneous equations. Inverse filtering is equivalent to getting an approximate solution for a set of simultaneous equations where the number of equations is higher than that of the unknowns. To obtain approximate solutions, one uses the normal equation that is employed in the linear regression of statistics although one's simultaneous equations are deterministic. The chapter also discusses the least square error method, signal representation by generalized discrete Fourier transforms, principal component analysis, and a pseudo-inverse method.