Homework 2 part C
What is a linear system?
A system is linear if:
1. The output of summing any two inputs togerther then sending the result through a system is equal to any two inputs sent through a system than added together.
2. Multiplying an input then sending it through the system equals the input sent through the system then multiplied by the constant.
proving this...
Example of a Linear System
First I will sum the inputs before the system
the system is y(t) = 5x(t).
a(t) = 3t
b(t) = 4t
c(t) = a(t) + b(t)
c(t) = 3t + 4t
y(t) = 5(3t + 4t) = 35t
If it works backwards it is linear:
y1(t) = 5(3t) = 15t
y2(t) = 5(4t) = 20t
y3(t) = 15t+20t = 35t
Both approaches yield the same result. Therefore it is linear.
Example of a non-linear System Given the system
y(t) = 10x(t) + 3:
x1(t) = t
y1(t) = 10t + 3
x2(t) = 4t
y2(t) = 40t + 3
y1(t) + y2(t) = 50t + 6
Do this backwards:
z(t) = x1(t)+y1(t)
z(t) = 5t
y[z(t)] = 50t + 6
60t+10 is not equal to 60t + 5
