Fourier Transforms

CHAPTER 2 VECTOR SPACES

Quantities in the physical world are modeled mathematically by functions of one or more independent variables. These functions are often called signals, particularly when they result from measurements or are purposely designed to convey information. An audio signal is a function of one independent variable, time. A photographic image is a function of two spatial variables. A video image is a function of one temporal and two spatial variables. The sound wave emanating from a loudspeaker is a function of time and three spatial variables.

For mathematical analysis, it is convenient to group signals into classes with common properties, such as: “all speech signals,” “all 256 × 256 pixel images,” “all signals with amplitude less than one volt,” “all continuous functions”. This chapter introduces several important signal classes and the mathematical structures that model them. With the right set of rules, these signals, be they finite-dimensional vectors, infinite sequences, or functions, can be collected into families that display behavior strikingly similar to physical vectors. These special families are called linear spaces, or vector spaces.

The properties of physical vectors are helpfully understood in geometric terms. The magnitude of a vector is expressed by its norm. Two vectors representing, say, force or velocity can be added to produce a new vector that combines the effects of the individual vectors. The dot product of two vectors expresses their ...