Digital Filters

by Richard W. Hamming

Overview

Digital signals occur in an increasing number of applications: in telephone communications; in radio, television, and stereo sound systems; and in spacecraft transmissions, to name just a few. This introductory text examines digital filtering, the processes of smoothing, predicting, differentiating, integrating, and separating signals, as well as the removal of noise from a signal. The processes bear particular relevance to computer applications, one of the focuses of this book.
Readers will find Hamming's analysis accessible and engaging, in recognition of the fact that many people with the strongest need for an understanding of digital filtering do not have a strong background in mathematics or electrical engineering. Thus, this book assumes only a knowledge of calculus and a smattering of statistics (reviewed in the text). Adopting the simplest, most direct mathematical tools, the author concentrates on linear signal processing; the main exceptions are the examination of round-off effects and a brief mention of Kalman filters.
This updated edition includes more material on the z-transform as well as additional examples and exercises for further reinforcement of each chapter's content. The result is an accessible, highly useful resource for the broad range of people working in the field of digital signal processing.

Product Details

ISBN-13:	9780486650883
Publisher:	Dover Publications
Publication date:	07/10/1997
Series:	Dover Civil and Mechanical Engineering
Edition description:	Revised
Pages:	304
Product dimensions:	5.50(w) x 8.50(h) x (d)

About the Author

Richard W. Hamming: The Computer Icon
Richard W. Hamming (1915-1998) was first a programmer of one of the earliest digital computers while assigned to the Manhattan Project in 1945, then for many years he worked at Bell Labs, and later at the Naval Postgraduate School in Monterey, California. He was a witty and iconoclastic mathematician and computer scientist whose work and influence still reverberates through the areas he was interested in and passionate about. Three of his long-lived books have been reprinted by Dover: Numerical Methods for Scientists and Engineers, 1987; Digital Filters, 1997; and Methods of Mathematics Applied to Calculus, Probability and Statistics, 2004.

In the Author's Own Words:
"The purpose of computing is insight, not numbers."

"There are wavelengths that people cannot see, there are sounds that people cannot hear, and maybe computers have thoughts that people cannot think."

"Whereas Newton could say, 'If I have seen a little farther than others, it is because I have stood on the shoulders of giants, I am forced to say, 'Today we stand on each other's feet.' Perhaps the central problem we face in all of computer science is how we are to get to the situation where we build on top of the work of others rather than redoing so much of it in a trivially different way."

"If you don't work on important problems, it's not likely that you'll do important work." -- Richard W. Hamming

Table of Contents

Preface to the third edition
1. Introduction
1.1 What is a digital filter?
1.2 Why should we care about digital filters?
1.3 How shall we treat the subject?
1.4 General-purpose versus special-purpose computers
1.5 Assumed statistical background
1.6 The distribution of a statistic
1.7 Noise amplification in a filter
1.8 Geometric progressions
2. The frequency approach
2.1 Introduction
2.2 Aliasing
2.3 The idea of an eigenfunction
2.4 Invariance under translation
2.5 Linear systems
2.6 The eigenfunctions of equally spaced sampling
2.7 Summary
3. Some classical applications
3.1 Introduction
3.2 Least-squares fitting of polynomials
3.3 Least-squares quadratics and quartics
3.4 Modified least squares
3.5 Differences and derivatives
3.6 More on smoothing: decibles
3.7 Missing data and interpolation
3.8 A class of nonrecursive smoothing filters
3.9 An example of how a filter works
3.10 Integration: recursive filters
3.11 Summary
4. Fourier series: continuous case
4.1 Need for the theory
4.2 Orthogonality
4.3 Formal expansions
4.4 Odd and even functions
4.5 Fourier series and least squares
4.6 Class of functions and rate of convergence
4.7 Convergence at a point of continuity
4.8 Convergence at a point of discontinuity
4.9 The complex Fourier series
4.10 The phase form of a Fourier series
5. Windows
5.1 Introduction
5.2 Generating new Fourier series: the convolution theorems
5.3 The Gibbs phenomenon
5.4 Lanczos smoothing: The sigma factors
5.5 The Gibbs phenomenon again
5.6 Modified Fourier series
5.7 The von Hann window: the raised cosine window
5.8 Hamming window: raised cosine with a platform
5.9 Review of windows
6. Design of nonrecursive filters
6.1 Introduction
6.2 A low-pass filter design
6.3 Continuous design methods: a review
6.4 A differentiation filter
6.5 Testing the differentiating filter on data
6.6 New filters from old ones: sharpening a filter
6.7 Bandpass differentiators
6.8 Midpoint formulas
7. Smooth nonrecursive filters
7.1 Objections to ripples in a transfer function
7.2 Smooth filters
7.3 Transforming to the Fourier series
7.4 Polynomial Processing in general
7.5 The design of a smooth filter
7.6 Smooth bandpass filters
8. The Fourier integral and the sampling theorem
8.1 Introduction
8.2 Summary of results
8.3 The Sampling theorem
8.4 The Fourier integral
8.5 Some transform pairs
8.6 Band-limited functions and the Sampling theorem
8.7 The convolution theorem
8.8 The effect of a finite sample size
8.9 Windows
8.10 The uncertainty principle
9. Kaiser windows and optimization
9.1 Windows
9.2 Review of Gibbs Phenomenon and the Rectangular window
9.3 The Kaiser window: I subscript 0-sinh window
9.4 Derivation of the Kaiser formulas
9.5 Design of a bandpass filter
9.6 Review of Kaiser window filter design
9.7 The same differentiator again
9.8 A particular case of differentiation
9.9 Optimizing a design
9.10 A Crude method of optimizing
10. The finite Fourier series
10.1 Introduction
10.2 Orthogonality
10.3 Relationship between the discrete and continuous expansions
10.4 The fast Fourier transform
10.5 Cosine expansions
10.6 Another method of design
10.7 Padding out zeros
11. The spectrum
11.1 Review
11.2 Finite sample effects
11.3 Aliasing
11.4 Computing the spectrum
11.5 Nonharmonic frequencies
11.6 Removal of the mean
11.7 The phase spectrum
11.8 Summary
12. Recursive filters
12.1 Why recursive filters?
12.2 Linear differential equation theory
12.3 Linear difference equations
12.4 Reduction to simpler form
12.5 Stability and the Z transformation
12.6 Butterworth Filters
12.7 A simple case of butterworth filter design
12.8 Removing the phase: two-way filters
13. Chebyshev approximation and Chebyshev filters
13.1 Introduction
13.2 Chebyshev polynomials
13.3 The Chebyshev Criterion
13.4 Chebyshev filters
13.5 Chebyshev filters, type 1
13.6 Chebyshev filters, type 2
13.7 Elliptic filters
13.8 Leveling an error curve
13.9 A Chebyshev identity
13.10 An example of the design of an integrator
13.11 Phase-free recursive filters
13.12 The transient
14. Miscellaneous
14.1 Types of Filter Design
14.2 Finite arithmetic effects
14.3 Recursive versus nonrecursive filters
14.4 Direct modeling
14.5 Decimation
14.6 Time-varying filters
14.7 References
Index

Editorial Reviews

Presents fundamental ideas and basic methods of designing simple nonrecursive and recursive filters. Uses simple, direct mathematical tools and includes an accurate (but not excessively rigorous) introduction to the necessary mathematics in each case. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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