**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Digital Filter Theory Summary

This section summarizes digital filter theory in a rather terse fashion suitable for review study.

linearity,time-invarianceand four basic representations of digital filters are defined: thedifference equation coefficients,impulse response, transfer function,andfrequency response. Next the concepts ofphase delayandgroup delayare defined. This material is a subset of that in [1].

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 .

Subsections