09. Linearity and Time-Invariance

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Linearity and Time-Invariance

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III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 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 $x_1(\ and for all constant gains $g$, we have

$\   $\ (40)
$\   $\ (41)

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

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


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

From now on, all filters discussed will be linear and time-invariant. For brevity, these will be referred to as LTI filters.

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