Difference between revisions of "HW2.C Bavorndej Chanyasak ECE301Fall2008mboutin" - Rhea

Latest revision as of 19:23, 11 September 2008

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.