A Linear system is a system that makes the result of 1. and 2. equal. 1. For any function x1(t) that goes into the system and is multiplied by A is added to a function x2(t) that goes into the system and is multiplied by B so that the added result is z(t)

    (X1(t) --> system --> *A)
+   (X2(t) --> system --> *B)
-----------------------------
z(t)

2. For any function x1(t) that is multiplied by A and added to any function x2(t) that is multiplied by B, of which then the whole goes into the system.

[Ax1(t) + Bx2(t)]--> system --> w(t)

So if w(t) = z(t) then the system is linear.

Example of a Linear System

Let the system be y(t) = 2x(t)+1

Let x1(t) = 1 Let x2(t) = n Let A = 1 Let B = 5

1.

    (1 --> system = (2(1)+1) = 3 --> 3*1 = 3)
+   (n --> system = (2(n)+1) = 2n+1 --> 5(2(n)+1) = 10n+5)
-----------------------------
 = 10n+8

2.

[1*1 + 5n = 5n+1] --> system = 2(5n+1)+1 = 10n+3

Note: this is supposed to be a linear system...but for the life of me, I can't figure out why the answers aren't equal. Got any ideas?

Example of Non Linear System

Lets say that the system is y(t) = e^x(t)

let x1(t) = 3 let x2(t) = 4 Let A = 2 Let B = 3

1. 3 --> system = e^3 --> *A = 2e^3
   4 --> system = e^4 --> *B = 3e^4
------------------------------------
z(t) = 2e^3 + 3e^4

2. [Ax1(t) + Bx2(t)] --> system --> w(t)

  [2*3 + 3*4 = 18] --> system = e^18

So e^18 is != 2e^3 + 3e^4 therefore this system is not linear.

Go back to : Homework 2_ECE301Fall2008mboutin

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