Linearity and Time-Invariance

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Linearity and Time-Invariance

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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

$\ $\(B.5)
$\ $\(B.6)

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.

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

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