Multirate and Wavelet Signal Processing
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About this ebook
This innovative and in-depth book integrates the well-developed theory and practical applications of one dimensional and multidimensional multirate signal processing. Using a rigorous mathematical framework, it carefully examines the fundamentals of this rapidly growing field. Areas covered include: basic building blocks of multirate signal processing; fundamentals of multidimensional multirate signal processing; multirate filter banks; lossless lattice structures; introduction to wavelet signal processing.
Multirate and Wavelet Signal Processing forms the basis for a graduate course in multirate signal processing. It includes an introduction to wavelet signal processing and emphasizes topics of ever-increasing importance for a wide range of applications. Concise and easy-to-read, this book is also a useful primer for professional engineers.
- Integrates the well-developed theory and practical applications of one-dimensional and multidimensional multirate signal processing
- Emphasizes topics of ever-increasing importance for a wide range of applications
- Written in a concise, easy-to-read style
- Uses relevant examples
- General mathematical formulation permits extensions of concepts to diverse applications, such as speech, imaging, video, and synthetic aperture radar
- Emphasizes key topics of the field, allowing the reader to make the most efficient use of time in learning the fundamentals of multirate
- Designed to be completely covered in a single semester or quarter
Bruce W. Suter
Bruce W. Suter received from the University of South Florida the BS and MS degrees in electrical engineering in 1972 and the Ph.D. in computer sciencein 1988. Since 1989, he has been with the Air Force Institute of Technology, where he is a professor in the Department of Electrical and Computer Engineering. He has also been with Honeywell, Inc., Litton Industries, and the University of Alabama at Birmingham. His current research interests include multirate signal processing, wavelet theory, time-frequency analysis, together with applications to special purpose computer architectures and multimedia communications systems.Dr. Suter is a member of Tau Beta Phi, Eta Kappa Nu, and a senior member of IEEE. In addition, he also serves as an Associate Editor of IEEE Transactions on Signal Processing.
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Book preview
Multirate and Wavelet Signal Processing - Bruce W. Suter
her.
Preface
Bruce W. Suter
Fairborn, Ohio February 9, 1997
The field of multirate and wavelet signal processing finds applications in speech and image compression, the digital audio and digital video industries, adaptive signal processing, and in many other applications.
The utilization of multirate techniques is becoming an indispensable tool of the electrical engineering profession. This point can be illustrated in three ways. First, if a performance specification is controlling the design of a particular system, that is, the performance specification exceeds the current state-of-art, then by converting the system to a multirate system, the overall system specification can be met with slower components. Secondly, if the dollar cost specification is controlling the design of a particular system, that is, the design of a competitive commercial system where bottom line cost is most important, then by converting the system to a multirate system, the overall system cost will be reduced through the utilization of slower, cheaper devices. Thirdly, if power consumption is controlling the design of a particular system, that is, the design of a hand-held system powered by a couple AA batteries, or possibly a satellite system, then by converting the system to a multirate system will reduce power consumption through the utilization of devices with slower switching speed, and as a result, lower power dissipation.
Wavelet transforms are closely related to filter banks. As such, a background in filter banks will make it easier for the reader to understand, design, and implement wavelet transforms.
Many of the most important applications, such as video compression, and many challenging research problems are in the area of multidimensional multirate. As such, multidimensional multirate is integrated throughout the book.
The focus of this book is to present a sound theoretical foundation by emphasizing the general principles of multirate. This book is self-contained for readers who have some prior exposure to linear algebra (at the level of Horn and Johnson’s Matrix Analysis) and multidimensional signal processing (at the level of Lim’s Two-Dimensional Signal and Image Processing or Dudgeon and Mersereau’s Multidimensional Digital Signal Processing). Moreover, this text will bring the reader to a point where he/she can read, understand, and appreciate the vast multirate literature.
The organization of this book is as follows. The first two chapters are devoted to basic multirate ideas including decimators, expanders, polyphase notation, etc. This presentation is first given for one-dimensional signals in Chapter 1 and then generalized to multidimensional signals in Chapter 2. The next two chapters deal with filter banks. Chapter 3 presents the theory of filter banks for both one-dimensional and multidimensional signals. Chapter 4 deals with lattice structures, an efficient implementation strategy for filter banks. Chapter 5 highlights an important application of multirate — the implementation of wavelets.
I would also like to take this opportunity to thank Professor Charles Chui for his enthusiasm about this project and for including this text in his distinguished wavelet series. The following people have provided very useful feedback during the writing of this book. They include: Bill Cowan, Tom Foltz, Jerry Gerace, Ying Huang, You Jang, Matt Kabrisky, Mark Oxley, Robert Parks, Juan Vasquez, and Dan Zahirniak.
Chapter 1
Multirate Signal Processing
1.1 Introduction
This chapter provides the basic concepts used in the study of multirate and wavelet signal processing. Some of the earliest contributions to the study of the fundamentals of multirate were due to Schafer and Rabiner[40], Meyer and Burrus[32], Oetken et al.[37], and Crochiere and Rabiner[10]. The idea of polyphase representation is a key concept throughout the development of this book. This nontrivial idea was first articulated by Bellanger et al.[3]. Much more recently, Evangalista[17] carefully examined another important idea – digital comb filters.
Many of the concepts developed in this chapter are also discussed in the other multirate texts by Crochiere and Rabiner[11], Fliege[19], Strang and Nguyen[46] and Vaidyanathan[49].
Section 1.2 presents a framework for multirate and it introduces two important representations for discrete signals. Section 1.3 introduces the basic building blocks. Section 1.4 provides ways to interchange the basic building blocks. Section 1.5 presents a filter bank example.
1.2 Foundations of multirate
First we will examine some sampling considerations and then present some basic transforms for analyzing signals.
1.2.1 Sampling considerations
Multirate is the study of time-varying systems. As such, the sampling rate will change at various points in time in an implementation. This will require us to vary the gain (magnitude) of filters in series with the time-varying building blocks so that the resulting gain is consistent with what one would expect if the sampling interval after the time-varying block hadbeen the original sampled frequency. Towards this end, let us analyze a train of impulses.
Theorem 1.2.1.1:
Proof: in a Fourier series. So,
where,
orequivalently,
Letτ = t - kT. Then,
Werecognize this as a sum of integrals with adjoining limits and simplify to
Hence,
.
Let F denote the Fourier transform. So that if x(t) is a signal, then
Letus examine the Fourier transform of an impulse train.
Theorem 1.2.1.2:
Proof: From the previous theorem,
Therefore,
Now, performing the Fourier transform of a sum of the product of the input x(t) and Dirac delta functions, which can be expressed as the convolution of the corresponding functions, produces