## Showing Linearity and Time Invariance, or Not

The filter is nonlinear and time invariant. The scaling property of linearity clearly fails since, scaling by gives the output signal , while . The filter is time invariant, however, because delaying by samples gives which is the same as .

The filter
is linear and *time varying*.
We can show linearity by setting the input to a linear combination of
two signals
, where and
are constants:

Thus, scaling and superposition are verified. The filter is time-varying, however, since the time-shifted output is which is not the same as the filter applied to a time-shifted input ( ). Note that in applying the time-invariance test, we time-shift the input signal only, not the coefficients.

The filter , where is any constant, is *nonlinear*
and time-invariant, in general. The condition for time invariance is
satisfied (in a degenerate way) because a constant signal equals all
shifts of itself. The constant filter *is* technically linear,
however, for , since
, even though the input
signal has no effect on the output signal at all.

Any filter of the form
is linear and
time-invariant. This is a special case of a *sliding linear
combination* (also called a *running weighted sum*, or
*moving average* when
).
All sliding linear combinations are linear,
and they are time-invariant as well when the coefficients (
) are constant with respect to time.

Sliding linear combinations may also include past *output*
samples as well (feedback terms). A simple example is any filter of
the form

Since linear combinations of linear combinations are linear combinations, we can use

*induction*to show linearity and time invariance of a constant sliding linear combination including feedback terms. In the case of this example, we have, for an input signal starting at time zero,

If the input signal is now replaced by , which is delayed by samples, then the output is for , followed by

or for all and . This establishes that each output sample from the filter of Eq.(4.7) can be expressed as a time-invariant linear combination of present and past samples.

**Next Section:**

Nonlinear Filter Example: Dynamic Range Compression

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Time-Invariant Filters