Time-Invariant Filters

Free Books Introduction to Digital 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, samples, then the output waveform is simply delayed by 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, for every input signal , we have

$\displaystyle {\cal L}_n\{$ SHIFT $\displaystyle _N\{x\}\}$	$\displaystyle =$	$\displaystyle {\cal L}_{n-N}\{x(\cdot)\}\;=\;y(n-N)$
	$\displaystyle =$	SHIFT $\displaystyle _{N,n}\{y\}, \protect$	(5.5)

where the

-sample shift operator is defined by

SHIFT $\displaystyle _{N,n}\{x\}\isdef x(n-N).$

On the signal level, we can write

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

Thus, SHIFT $_N\{x\}$ denotes the waveform $x(\cdot)$ shifted right (delayed) by

samples. The most common notation in the literature for SHIFT $_N\{x\}$ is

, but this can be misunderstood (if

is not interpreted as ` $\cdot$ '), so it will be avoided here. Note that Eq. $\,$ (4.5) can be written on the waveform level instead of the sample level as