Space and Linearity

Abstract

The ability to see the geometric properties of objects in space helps to visualize important properties of digital signals. It is only necessary to find the correspondence between the signal property and the geometric object. A starting point is the analytic geometry of the line, the plane, and space. However, most of the notions of geometry such as space, distance, angle, orientation, and motion can be defined much more generally,

Digital signals representing sounds and images are modeled by points in some of these generalized spaces, and many common transformations of such signals are easily described as geometric operations on those points. For example, points in space may be added together or multiplied by real numbers, which correspond respectively to mixing signals or amplifying them. The results are variously called linear combinations, superpositions, or linear transformations.