In this chapter we discuss some basic concepts of Hilbert space and the related operations and properties as the mathematical foundation for the topics of the subsequent chapters. Specifically, based on the concept of unitary transformation in a Hilbert space, all of the unitary transform methods to be specifically considered in the following chapters can be treated from a unified point of view: they are just a set of different rotations of the standard basis of the Hilbert space in which a given signal, as a vector, resides. By such a rotation the signal can be better represented in the sense that the various signal processing needs, such as noise filtering, information extraction and data compression, can all be carried out more effectively and efficiently.

Inner product space

Vector space

In our future discussion, any signal, either a continuous one represented as a time function x(t), or a discrete one represented as a vector x = […, x[n], …]T, will be considered as a vector in a vector space, which is just a generalization of the familiar concept of N-dimensional (N-D) space, formally defined as below.

Definition 2.1.A vector space is a set v with two operations of addition and scalar multiplication defined for its members, referred to as vectors.