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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
- 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
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