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.

Why wavelet?

Short-time Fourier transform and Gabor transform

In Chapter 3, we learned that a signal can be represented as either a time function x(t) as the amplitude of the signal at any given moment t, or, alternatively and equivalently, as a spectrum X(f) = F[x(t)] representing the magnitude and phase of the frequency component at any given frequency f. However, no information in terms of the frequency contents is explicitly available in the time domain, and no information in terms of the temporal characteristics of the signal is explicitly available in the frequency domain. In this sense, neither x(t) in the time domain nor X(f) in the frequency domain provides complete description of the signal. In other words, we can have either temporal or spectral locality regarding the information contained in the signal, but never both at the same time.

To address this dilemma, the short-time Fourier transform (STFT), also called windowed Fourier transform, can be used. The signal x(t) to be analyzed is first truncated by a window function w(t) before it is Fourier transformed to the frequency domain. As all frequency components in the spectrum are known to be contained in the signal segment inside this particular time window, certain temporal locality in the frequency domain is achieved.

What is the book all about?

This is Euclid's definition for “perpendicular”, which is synonymous with the word “orthogonal” used in the title of this book. Although the meaning of this word has been generalized since Euclid's time to describe the relationship between two functions as well as two vectors, as what we will be mostly concerned with in this book, they are essentially no different from two perpendicular straight lines, as discussed by Euclid some 23 centuries ago.

Orthogonality is of important significance not only in geometry and mathematics, but also in science and engineering in general, and in data processing and analysis in particular. This book is about a set of mathematical and computational methods, known collectively as the orthogonal transforms, that enables us to take advantage of the orthogonal axes of the space in which the data reside. As we will see throughout the book, such orthogonality is a much desired property that can keep things untangled and nicely separated for ease of manipulation, and an orthogonal transform can rotate a signal, represented as a vector in a Euclidean space, or more generally Hilbert space, in such a way that the signal components tend to become, approximately or accurately, orthogonal to each other.

INTRODUCTION

The final shapes of most mechanical parts are obtained by machining operations. Bulk deformation processes, such as forging and rolling, and casting processes are mostly followed by a series of metal-removing operations to achieve parts with desired shapes, dimensions, and surface finish quality. The machining operations can be classified under two major categories: cutting and grinding processes. The cutting operations are used to remove material from the blank. The subsequent grinding operations provide a good surface finish and precision dimensions to the part. The most common cutting operations are turning, milling, and drilling followed by special operations such as boring, broaching, hobing, shaping, and form cutting. However, all metal cutting operations share the same principles of mechanics, but their geometry and kinematics may differ from each other. The mechanics of cutting and the specific analysis for a variety of machining operations and tool geometries are not widely covered in this text. Instead, a brief introduction to the fundamentals of cutting mechanics and a comprehensive discussion of the mechanics of milling operations are presented. Readers are referred to established metal cutting texts authored by Armarego and Brown [25], Shaw [96], and Oxley [83] for detailed treatment of the machining processes.

MECHANICS OF ORTHOGONAL CUTTING

Although the most common cutting operations are three-dimensional and geometrically complex, the simple case of two-dimensional orthogonal cutting is used to explain the general mechanics of metal removal.