Difference between revisions of "HW2-C Joseph Mazzei ECE301Fall2008mboutin" - Rhea

Latest revision as of 10:22, 12 September 2008

If Z(t) and W(t) in the following are equal the system is linear.

Linear:

An example of an linear function is $\ y(t) = 3x(t + 8)$

because the result of the 1st method above yields $\ 3ax(t + 8) + 3bx(t + 8)$

and the result of the 2nd method above yields $\ 3[ax(t + 8) + bx(t + 8)]$ . Because they yield the same result the system is linear.

Non-Linear:

An example of a non-linear function is $\ y(t) = e^{x(t)}$

because the result of the 1st method above yields $\ e^{ax(t)} + e^{bx(t)}$

and the result of the 2nd method above yields $\ e^{ax(t) + bx(t)}$ , which is not equal to the first result.

@@ Line 15: / Line 15: @@
 Linear:
-<math>\ S_{1} = 2x(t + 3) + x(t - 8)</math>
+An example of an linear function is <math>\ y(t) = 3x(t + 8)</math>
-<math>\ S_{2} = x(t - t_{0})</math>
+because the result of the 1st method above yields <math>\ 3ax(t + 8) + 3bx(t + 8)</math>
-<math>\ S_{1} \rightarrow S_{2} = 2x(t - t_{0} + 3) + x(t - t_{0} - 8)</math>
-<math>\ S_{2} \rightarrow S_{1} = 2x(t - t_{0} + 3) + x(t - t_{0} - 8)</math>
-Since the results are the same the system is time invariable.
+and the result of the 2nd method above yields <math>\ 3[ax(t + 8) + bx(t + 8)]</math>. Because they yield the same result the system is linear.
 Non-Linear:
-<math>\ S_{1} = x(-t + 3) - x(-t - 8)</math>
+An example of a non-linear function is <math>\ y(t) = e^{x(t)} </math>
-<math>\ S_{2} = x(t - t_{0})</math>
-<math>\ S_{1} \rightarrow S_{2} = x(-t + t_{0} + 3) - x(-t + t_{0} - 8)</math>
-<math>\ S_{2} \rightarrow S_{1} = x(-t - t_{0} + 3) - x(-t - t_{0} - 8)</math>
+because the result of the 1st method above yields <math>\ e^{ax(t)} + e^{bx(t)}</math>
-Since the results are different they system is time variant.
+and the result of the 2nd method above yields <math>\ e^{ax(t) + bx(t)}</math>, which is not equal to the first result.