## 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:

*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,

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Nonlinear Filter Example: Dynamic Range Compression

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Time-Invariant Filters