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Outline

Below is an overview of the chapters.

  1. 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.

  2. 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.

  3. 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.

  4. Linearity and Time-Invariance -- mathematical foundations of digital filter analysis, implications of linearity and time invariance, and various technical terms relating to digital filters.

  5. Time Domain Filter Representations -- difference equation, signal flow graphs, direct-form I, direct-form II, impulse response, the convolution representation, and FIR filters.

  6. Transfer Function Analysis -- the transfer function is a frequency-domain representation of a digital filter obtained by taking the z transform of the difference equation.

  7. 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 $ z$ 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.

  8. 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 $ z$ plane.

  9. Implementation Structures -- four direct-form implementations for digital filters, and series/parallel decompositions.

  10. 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.

  11. 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.

  12. 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 plugin generation).


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA