Linearity, Causality and Time-Invariance of a System¶

The notion of a system is central in digital communications and particularly system's theory. Abstractly, a system is defined as something that takes an input signal and produces an output signal by some transformation rule $Tr$. $$y(t)=Tr\{x(t)\}$$

Many relations in the real world can actually be understood as a system. Some examples include:

You press a key on your keyboard, and the corresponding letter appears on your screen. What happens if you press two letters at the same time? Is this system "linear"?

You speak into your microphone, and it converts your voice into electrical current. Hopefully this system does not introduce a lot of distortion.

You inflate the tire of your bike. It responds with the pressure in the tire. The pressure can be seen as the summation of all the air that has flown into and out of the tire.

Let's take a more abstract example: A system can amplify the input signal, by doubling its amplitude: $$y(t)=2x(t)$$ Let us illustrate this very simple system with some code. As an example, we take the input signal $x(t)=\sin(t)$. Then, we can straight-forwardly implement the system as taking a function (a signal) and returning another time function (another signal):