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 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

On the signal level, we can write

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.(4.5) can be written on the waveform level instead of the sample level as

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.