# What is a non causal system?

I am stydying control systems, and I am trying to understand better the topic of causality.

I know that a system is causal if the actual state depends only on the past states and on the present state, so I can represent a causal system for example as:

$$\y(5)=x(4)\$$

And also I could represent a transfer function of a causal system as for example:

$$\\frac{1}{s+1}e^{-s}\$$

which has as step response:

which is not related to the previous example in state space, but I wrote this just as an example.

Now, I would like to understand better the behavior of a non-causal system. As far as I know, a non-causal system is not realizable, since it is a system in which the actual state depends on the future states, and this is impossible.

But I can't understand well this concept of non causality. For example what is an example of transfer function of a non causal system and its step response or frequency response?

• If the current output depends on one or more future inputs, the system is non-causal. For example, a simple three-point moving average of a data set, where the complete data set is known a-priori, may be defined: $y[k]=\frac{1}{3}\left( x[k-1]+x[k]+x[k+1]\right)$, where $x[k]$ are the input data, and $y[k]$ are the output data. – Chu Feb 15 '20 at 11:15