**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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-invarianceand four basic representations of digital filters are defined: thedifference equation coefficients, impulse response, transfer function,andfrequency response. Next the concepts ofphase delay,group delay,polesandzeros, and filterstabilityare defined. This elementary material was taken from course notes for a class given at Stanford by the author in 1979.

Definition.A realsignalis defined as any real-valued function of the integers. Similarly a complex signal is any complex-valued function of the integers.

Definition.A realfilteris defined as any real-valued functional of a signal for each integer . We express the input-output relation of the filter by

where is the entire input signal, and is the output at time .