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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, for every input signal $ x$, 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 $ N$-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 $ 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$'), so it will be avoided here. Note that Eq.$ \,$(4.5) can be written on the waveform level instead of the sample level as

$\displaystyle {\cal L}\{$SHIFT$\displaystyle _N\{x\}\}=$SHIFT$\displaystyle _N\{{\cal L}\{x\}\}=$SHIFT$\displaystyle _N\{y\}. \protect$ (5.6)

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