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 topics taken up
later are introduced in a simple setting.
MatlabFilter 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
digital 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
equation.
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.
Elementary and Important Digital Filters in Audio -- analysis of commonly used filters such as the one-zero, one-pole,
two-pole, two-zero, complex resonator, biquad, allpass, equalizers,
shelving filters, time-varying sections, constant-gain resonator, and
the dc blocker.
Minimum Phase Digital Filters -- minimum phase is the most ``natural'' phase response for a recursive
digital filter.
Appendices -- elementary discussion of
signal representation, complex and trigonometric identities, closure
of sinusoids under addition, proof of the convolution theorem for z transform
s, introduction to Laplace transform analysis, allpass filters,
state-space models, elementary digital filter design, links to
on-line resources, and software examples and utilities in matlab and
C++.
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.
