In everyday terms, the fact that a filter islinear means 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.
for all . These two conditions are a mathematical statement of the previous definition. For rational, property (2) implies (1).
where is understood to denote the waveform shifted right (or delayed) by samples.
Linear, time-invariant filters are often referred to as LTI filters.