This book was written for my introductory course in digital audio signal processing, which I have given at the Center for Computer Research in Music and Acoustics (CCRMA) since 1984. The course was created primarily as a first course in digital signal processing for entering Music Ph.D. students in the Computer Based Music Theory and Acoustics program. Due to the nature of CCRMA research, this book will emphasize audio and music applications, although the material on the subject of digital filters itself is not specific to audio or music. A recommended co-requisite is , which contains a more detailed development of the mathematics of signals and spectra in the discrete-time case. A shorter precursor to this book was published in .
Below is an overview of the chapters.
- The Simplest Lowpass Filter -- a thorough analysis of an extremely simple digital filter using
high-school level math (trigonometry) followed by a simpler but more
advanced approach using complex variables. Important later topics
are introduced in a simple setting.
- Matlab Filter Analysis -- a thorough analysis of the same simple digital filter analyzed in
Chapter 1, but now using the matlab programming language. Important
computational tools are introduced while the study of filter
theory is hopefully being motivated.
- Analysis of Digital Comb Filter -- a thorough analysis and display of an example digital comb filter of
practical complexity using more advanced methods, both mathematically
and in software. The intent is to illustrate the mechanics of
practical digital filter analysis and to motivate mastery of the theory
presented in later chapters.
- Linearity and Time-Invariance -- mathematical foundations of digital filter analysis, implications of
linearity and time invariance, and various technical terms relating to
- Time Domain Filter Representations -- difference equation, signal flow graphs, direct-form I, direct-form II,
impulse response, the convolution representation, and FIR filters.
- Transfer Function Analysis -- the transfer function is a frequency-domain representation of a
digital filter obtained by taking the z transform of the difference
- Frequency Response Analysis -- the frequency response is a frequency-domain representation of a
digital filter obtained by evaluating the transfer function on the
unit circle in the plane. The magnitude and phase of the
frequency response give the amplitude response and phase response,
respectively. These functions give the gain and delay of the filter
at each frequency. The phase response can be converted to the more
intuitive phase delay and group delay.
- Pole-Zero Analysis -- poles and zeros provide another frequency-domain representation
obtained by factoring the transfer function into first-order
terms. The amplitude response and phase response can be quickly
estimated by hand (or mentally) using a graphical construction based
on the poles and zeros. A digital filter is stable if and only if its
poles lie inside the unit circle in the plane.
- Implementation Structures -- four direct-form implementations
for digital filters, and series/parallel decompositions.
- Filters Preserving Phase -- zero-phase and linear-phase
digital filters. Such filters largely preserve the shape of a signal
in the time domain by delaying all frequency components equally.
- Minimum-Phase Filters -- minimum-phase digital filters,
desired phase for audio filters, and creating minimum phase from
spectral magnitude. Minimum phase gives ``minimum delay'' for a
useful class of filters.
- Appendices -- elementary discussion of
signal representation, complex and trig identities, closure
of sinusoids under addition,
elementary digital filters, equalizers, time-varying resonators, the
dc blocker, allpass filters, introduction to Laplace transform
analysis, analog filters, state-space models, elementary filter
design, links to on-line resources, and software examples in
the matlab, C++, and Faust programming languages (including automatic
- MATHEMATICS OF THE DISCRETE FOURIER TRANSFORM
All about the DFT formula and its constituents, with frequent references to audio applications.
- INTRODUCTION TO DIGITAL
A gentle introduction to the analysis and implementation of digital filters, with particular emphasis on audio applications.
- PHYSICAL AUDIO SIGNAL PROCESSING
Efficient computational physical models for delay effects and virtual acoustic musical instruments.
- SPECTRAL AUDIO SIGNAL PROCESSING
Analysis, processing, and synthesis of audio signals in terms of spectral representations computed using the Fast Fourier Transform (FFT).
Figure 1 illustrates the dependencies. A solid line indicates a strong dependence, while a dotted line indicates a weaker (optional) dependence. The student is expected to pick up elementary physics  and programming skills [82,15] elsewhere. In all books, the main chapters contain approximately what is covered in class, while the appendices provide both elementary background material and additional advanced topics.
With respect to the previously published precursor of this book , I would like to again express appreciation to John Strawn for his thorough editing assistance, and to Andy Schloss, David Jaffe, Andy Moorer, Janet Coursey, and Ken Shoemake for their helpful proofreading and suggestions. More recently, thanks to Vivian Woo and Matt Wright for especially helpful reviewing assistance, and to Mark Cartwright, Ryan Cassidy, Humane Chan, Nick Dargahi, David Gelbart, Peter Howard, Thomas Icking, Tim Janik, Christian Keil, Moonseok Kim, Arvindh Krishnaswamy, Andrew Leary, Dong In Lee, Poliang Lin, Gautham Mysore, Jim Murphy, Juhan Nam, John Nolting, Carlos Pita, Mathew Romaine, Volker Schatz, Andrew Simper, Lee Soulsby, Sook Young Won, and Larry Wu, for reporting errata they encountered in the earlier draft versions of this book.
The support of the Fannie and John Hertz Foundation during my graduate student years--when  was written--is gratefully acknowledged. Finally, the present version would not exist at all without the support of my wife Carol and son Harrison (who made sure I took plenty of breaks along the way ).
Despite best efforts, errata will be almost surely be discovered over time. See http://books.w3k.org/filters1p1/ for known errata, clarifications, and other information pertaining to this book.
The author and publisher make no warranties, express or implied, regarding the contents of this book.
The Simplest Lowpass Filter