**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Linearity and Time-Invariance

In everyday terms, the fact that a filter is

linearmeans simply that

- the amplitude of the output is proportional to the amplitude of the input, and
- 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 belinearif for any pair of signals and for all constant gains , we have

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 betime-invariantif

where is understood to denote the waveform shifted right (or delayed) by samples.From now on, all filters discussed will be linear and time-invariant. For brevity, these will be referred to as

LTI filters.