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
Showing Linearity and Time Invariance, or Not