A time-invariant system is a system that has a fixed output over a certain time. In other words, the time-shifted output signal must correspond to the time-shifted input signal.

To prove this property, let:


$ y(t) = sin[x(t)]\, $


for a particular system, then choosing some arbitraty inputs:


$ y_1(t) = sin[x_1(t)]\, $


and suppose


$ x_2(t) = x_1(t - t_0)\, $


Therefore


$ y_2(t) = sin[x_2(t)] = sin[x_1(t - t_0)]\, $


Equivalently


$ y_1(t - t_0) = sin[x_1(t - t_0)]\, $


It is clear that $ y_2(t) = y-1(t - t_0) $, so this system is time invariant.

As a prove for time-variant system, let:


$ y(t) = tx(t)\, $


Again, choosing arbitrary inputs


$ y_1(t) = tx_1(t)\, $


and


$ x_2(t) = x_1(t - t_0)\, $


The output would be


$ y_2(t) = tx_2(t) = tx_1(t - t_0)\, $


But


$ y_1(t - t_0) = (t - t_0)x(t - t_0)\, $


As we can see, $ y_2(t) $ is not equal to $ y_1(t - t_0) $, and therefore is time-variant.

Note: The examples shown here were taken from Signals & Systems Second edition by Alan V. Oppenheim and Alan S. Willsky pg.51

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett