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

Thus,
SHIFT denotes the waveform shifted right
(delayed) by samples. The most common notation in the literature
for
SHIFT is , but this can be misunderstood (if
is not interpreted as `'), so it will be avoided here.
Note that Eq. (4.7) can be written on the waveform level instead
of the sample level as