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
