Showing Linearity and Time Invariance Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

## 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 , while . The filter is time invariant, however, since 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). 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

 (5.9)

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. (4.9) can be expressed as a time-invariant linear combination of present and past samples.

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]