# Linearity and Time-Invariance

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Linearity and Time-Invariance

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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## Linearity and Time-Invariance

In everyday terms, the fact that a filter islinear means simply that

1. the amplitude of the output is proportional to the amplitude of the input, and
2. when two signals are added together and fed to the filter, the filter output is the same as if one had put each signal through the filter separately and then added the outputs.

Definition. A filter is said to be linear if for any pair of signals and for all constant gains , we have

 (B.5) (B.6)

for all . These two conditions are a mathematical statement of the previous definition. For rational, property (2) implies (1).

Definition. A filter is said to be time-invariant if

where is understood to denote the waveform shifted right (or delayed) by samples.

Linear, time-invariant filters are often referred to as LTI filters.

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