NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Introduction to Digital Filter Theory
In this section, linearity, time-invariance and four basic representations of digital filters are defined: thedifference equation coefficients, impulse response, transfer function, and frequency response. Next the concepts ofphase delay, group delay, poles and zeros, and filter stability are defined. This elementary material was taken from course notes for a class given at Stanford by the author in 1979.
Definition. A real signal is defined as any real-valued function of the integers. Similarly a complex signal is any complex-valued function of the integers.
Definition. A real filter
is defined as any real-valued functional of a signal for each integer
. We express the input-output relation of the filter by
whereis the entire input signal, and
is the output at time
