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 $x(\cdot)$ as the symbol for the filter input. What we want to say is that if the input signal is delayed (shifted) by, say, $N$ samples, then the output waveform is simply delayed by $N$ samples and unchanged otherwise. Thus $y(\cdot)$, the output waveform from a time-invariant filter, merely shifts forward or backward in time as the input waveform $x(\cdot)$ is shifted forward or backward in time.

Definition. A digital filter ${\cal L}_n$ is said to be time-invariant if

$${\cal L}_n\{ _N\{x\}\} = {\cal L}_{n-N}\{x(\cdot)\}=y(n-N)$$ (5.6)

$$= _{N,n}\{y\}, \protect$$ (5.7)

where the $N$-sample shift operator is defined by

   SHIFT$$_{N,n}\{x\}\isdef x(n-N).$$

On the signal level, we can write

   SHIFT$$_N\{x\} \isdef x(\cdot-N).$$

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

$${\cal L}\{ _N\{x\}\}= _N\{{\cal L}\{x\}\}= _N\{y\}. \protect$$ (5.8)