Difference between revisions of "HW2-C Scott Hamilton ECE301Fall2008mboutin" - Rhea

Revision as of 06:26, 12 September 2008

A linear is system is a system that given two valid inputs:

$ x_1(t) $

$ x_2(t) $

with respective outputs:

y_1(t) = H { x_1(t) }

y_2(t) = H { x_2(t) }

will satisfy the equation

\alpha y_1(t) + \beta y_2(t) = H { \alpha x_1(t) + \beta x_2(t) }

for any $\alpha$ and $\beta$ .

@@ Line 1: / Line 1: @@
 ==Problem 4==
-A general deterministic system can be described by operator <math>H</math> that maps an input <math>x(t)</math> as a function of <math>t</math> to an output <math>y(t)</math>.
+A linear is system is a system that given two valid inputs:
+:<math>x_1(t) </math>
-Given two valid inputs
+:<math>x_2(t) </math>
-:<math>x_1(t) \,</math>
+with respective outputs:
-:<math>x_2(t) \,</math>
+:<math>y_1(t) = H { x_1(t) } </math>
-as well as their respective outputs
+:<math>y_2(t) = H { x_2(t) } </math>
-:<math>y_1(t) = H \left \{ x_1(t) \right \} </math>
+will satisfy the equation
-:<math>y_2(t) = H \left \{ x_2(t) \right \} </math>
+:<math>\alpha y_1(t) + \beta y_2(t) = H { \alpha x_1(t) + \beta x_2(t) } </math>
-then a linear system must satisfy
+for any  <math>\alpha </math> and <math>\beta </math>.
-:<math>\alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math>
-for any [[scalar (mathematics)_ECE301Fall2008mboutin|scalar]] values <math>\alpha \,</math> and <math>\beta \,</math>.
-<!-- Insert picture dipicting the superposition and scaling properties -->