A linear system is a system that an output of a certain signal is the sum of all the input signals. This is exactly the same as the property called superposition. ie:
- $ ax_1(t) + bx_2(t) > ay_1(t) + by_2(t)\, $
This definition also holds true for DT signals.
For example, to prove that a system is linear, suppose that a system with output $ y(t) $ and input $ x(t) $ are related by
- $ y(t) = tx(t)\, $
Choosing arbitrary signals, we have:
- $ x_1(t) > y_1(t) = tx_1(t)\, $
- $ x_2(t) > y_2(t) = tx_2(t)\, $
and let
- $ x_3(t) = ax_1(t) + bx_2(t)\, $
Therefore,
- $ y_3(t) = tx_3(t)\, $
- $ y_3(t) = t(ax_1(t) + bx_2(t))\, $
- $ y_3(t) = atx_1(t) + btx_2(t))\, $
- $ y_3(t) = ay_1(t) + by_2(t)\, $
Which correspond to the definition described above.
On the other hand, to prove that a system is non-linear, let's assume we have:
- $ y(t) = x^2(t)\, $
Again, choosing arbitrary signals:
- $ x_1(t) > y_1(t) = x_1^2(t)\, $
- $ x_2(t) > y_2(t) = x_2^2(t)\, $
and let:
- $ y_3(t) = x_3^2(t)\, $
- $ y_3(t) = (ax_1(t) + bx_2(t))^2\, $
- $ y_3(t) = a^2x_1^2(t) + b^2x_2^2(t) + 2abx_1(t)x_2(t)\, $
- $ y_3(t) = a^2y_1(t) + b^2y_2(t) + 2abx_1(t)x_2(t)\, $
As illustrated above, one can specify that $ y_3(t) $ is not the same as $ ay_1(t) + by_2(t) $, and therefore the system is not linear.
Note: Both of the examples above were taken from Signals & Systems, second edition by Alan V. Oppenheim and Alan S. Willsky pg. 54.
