NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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-invariance and four basic representations of digital filters are defined: the difference equation coefficients,impulse response, transfer function, and frequency response. Next the concepts of phase delay and group delay are defined. This material is a subset of that in [1].

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

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

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Subsections

• Linearity and Time-Invariance
• Difference Equation
• Convolution Representation of LTI Filters
• Frequency Response
• Phase Delay and Group Delay

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