## Time-Invariant Filters

In plain terms, a *time-invariant
filter* (or *shift-invariant
filter*) is one which performs the
*same operation at all times*. It is awkward to express this
mathematically by restrictions on Eq.(4.2) because of the use of
as the symbol for the filter input. What we want to say is
that if the input signal is delayed (shifted) by, say, samples,
then the output waveform is simply delayed by samples and
unchanged otherwise. Thus , the output waveform from a
time-invariant filter, merely *shifts* forward or backward in
time as the input waveform is shifted forward or backward
in time.

**Definition. **A digital filter
is said to be
*time-invariant*
if, for every input signal , we have

where the -sample

*shift operator*is defined by

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