Digital Filters

Richard W. Hamming

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486650883
• eBook ISBN: 9780486319247
• Topic: Technology & Engineering
• Subtopic: Civil Engineering
• Index: Technology & Engineering

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.

1

Introduction

1.1 WHAT IS A DIGITAL FILTER?

In our current technical society we often measure a continuously varying quantity. Some examples include blood pressure, earthquake displacements, voltage from a voice signal in a telephone conversation, brightness of a star, population of a city, waves falling on a beach, and the probability of death. All these measurements vary with time; we regard them as functions of time: u(t) in mathematical notation. And we may be concerned with blood pressure measurements from moment to moment or from year to year. Furthermore, we may be concerned with functions whose independent variable is not time, for example the number of particles that decay in a physics experiment as a function of the energy of the emitted particle. Usually these variables can be regarded as varying continuously (analog signals) even if, as with the population of a city, a bacterial colony, or the number of particles in the physics experiment, the number being measured must change by unit amounts.

For technical reasons, instead of the signal u(t), we usually record equally spaced samples u_n of the function u(t). The famous sampling theorem, which will be discussed in Chapter 8, gives the conditions on the signal that justify this sampling process. Moreover, when the samples are taken they are not recorded with infinite precision but are rounded off (sometimes chopped off) to comparatively few digits (see Figure 1.1-1). This procedure is often called quantizing the samples. It is these quantized samples that are available for the processing that we do. We do the processing in order to understand what the function samples u_n reveal about the underlying phenomena that gave rise to the observations, and digital filters are the main processing tool.

FIGURE 1.1-1 SAMPLING AND QUANTIZATION OF A SIGNAL
digital filters are the main processing tool

It is necessary to emphasize that the samples are assumed to be equally spaced; any error or noise is in the measurements u_n. Fortunately, this assumption is approximately true in most applications.

Suppose that the sequence of numbers {u_n} is such a set of equally spaced measurements of some quantity u(t), where n is an integer and t is a continuous variable. Typically, t represents time, but not necessarily so. We are using the notation u_n = u(n). The simplest kinds of filters are the nonrecursive filters; they are defined by the linear formula

The coefficients c_k are the constants of the filter, the u_n-k are the input data, and the y_n are the outputs. Figure 1.1-2 shows how this formula is computed. Imagine two strips of paper. On the first strip, written one below the other, are the data values u_n-k. On the second strip, with the values written in the reverse direction (from bottom to top), are the filter coefficients c_k. The zero subscript of one is opposite the n subscript value of the other (either way). The output y_n is the sum of all the products c_ku_n-k. Having computed one value, one strip, say the coefficient strip, is moved one space down, and the new set of products is computed to give the new output y_n+1. Each output is the result of adding all the products formed from the proper displacement between the two zero-subscripted terms. In the computer, of course, it is the data that is “run past” the coefficient array {c_k}.

This process is basic and is called a convolution of the data with the coefficients. It does not matter which strip is written in the rever...

Table of contents

• Contents
• Preface to the Third Edition
• 1 Introduction
• 2 The Frequency Approach
• 3 Some Classical Applications
• 4 Fourier Series: Continuous Case
• 5 Windows
• 6 Design of Nonrecursive Filters
• 7 Smooth Nonrecursive Filters
• 8 The Fourier Integral and the Sampling Theorem
• 9 Kaiser Windows And Optimization
• 10 The Finite Fourier Series
• 11 The Spectrum
• 12 Recursive Filters
• 13 Chebyshev Approximation And Chebyshev Filters
• 14 Miscellaneous
• Index