BME 343 - LTIC Systems
When I was trying to understand Linear, Time invariant continuous systems, I was confused by the what is considered the input signal, output signal, and system in a mathematical equation.
As a quick exercise, try to define the input, output, and system when you bounce a ball:
For this exercise, I defined the following:
Input: The force first exerted on the ball
Output: The height of the ball
System: All the variable the influence the ball such as gravity, height, friction, elasticity, and your actions
We have two models to describe this signal:
1. Physic based equations:
We can tie the behavior of the ball to physical equations we know. Because the world is complex, the equations will get complicated such as using second degree differential equation.
That sucks. Worse is that if we wanted to see the response (height of ball) of the system to different input (force onto ball), then we have to go through all the calculations again. Luckily, we are lazy so we have a smarter way of doing this:
2. Systems
Instead of thinking of all the different parameters in the system, we can generalized the input as f(t), where f(t) is the force on the ball including how the system (gravity, height, friction, and you) influence on it.
Then we can just say f(t) = h(t), where after we apply an input (force on the ball), we get the response (height).
(Note: If the notation “=” is confusing, think of it as ->).
Now that we defined our system, we can describe it.
LTIC Systems
As you know, LTIC means it has behavior that is both linear and time invariant. Let’s break that down.
Linear = Homogeneity & Superposition
Linearity is mathematically shown as showing homogeneity and superposition, so we can break this down even more.
Homogeneity:
If the system displays homogeneity, then whatever we multiply the input, the response will be multiplied by the same factor.
ax(t) = ay(t).
How can we see that in our bouncing ball system, assuming it shows homogeneity? Think about it for a moment…
If we bounced the ball with twice the force, then the height would double. Pretty intuitive right?
If f(t) = h(t), then 2f(t) = 2h(t)
Superposition:
If the system displays superposition, then the sum of the inputs would equal the sum of the outputs.
x1+x2 = y1 + y2.
How does this show in our bouncing ball system, assuming it shows superposition? Think about it for a moment…
First, we bounce the ball with force f1(t) which gives us h1(t)
Then, we bounce the ball with force f2(t) which gives us h2(t)
Since this system shows superposition, then we know adding the forces f1(t) and f2(t) will give us a height that is the sum of h1(t) and h2(t)
f1(t) + f2(t) = h1(t) + h2(t)
Note: Though similar, an important distinction between homogeneity and superposition is that homogeneity can be shown with one input/output pair, but superposition requires at least two.
Linear Systems:
So if we have two input/output that shows both homogeneity:
af1(t) = ah1(t)
bf2(t) = bh2(t)
and they show superposition with each other:
f1(t) + f2(t) = h1(t) + h2(t)
Then we can combine these characteristics to show a linear system:
af1(t) + bf2(t) = ah1(t) + bh2(t)
If we doubled the first force and quartered the second force, we know that response (height) would be the sum of the doubled first height and quartered second height.
Time Invariant:
If the system f(t) = h(t) is time invariant, then f(t-to) = h(t-to).
How would this be shown with our bouncing ball system? Think about it for a moment…
If you bounce the ball at 2 seconds later, then it will reach the height 2 seconds later.
f(t-2) = h(t-2)
In conclusion: When a system is both linear and time invariant, then it can be described as an LTIC system