2.6 Causality

System Causality

A system $H$ is causal if its behavior does not rely on future inputs. In other words, the output $y(n)$ depends only on inputs $x(k)$ where $k\le n$. Conversely, if $H$ contains a term where $k>n$  (e.g., $k=n+1$ s.t. $x(k)=x(n+1)$), the system would rely on future inputs and therefore cannot be considered causal.

Additionally, a discrete-time LTI system is causal if and only if $h(n)=0$ $\forall n<0$, where $n\in \mathbb{Z}$, and a continuous-time LTI system is causal if and only if $h(t)=0$ $\forall t<0$, where $t\in \mathbb{R}$.

Proof

We can prove causality for discrete-time LTI systems very quickly below:

1. $h(n)=0(\forall n<0)⟹\text{System is causal.}$
   We can simplify the convolution of a causal system such that its sum begins at 0:
   $y(n)={\sum }_{k=0}^{\infty }x(n-k)h(k).$
   Since $k=0$ is the initial value of the convolution, this means that there are no inputs $x(n-k)$ in $y(n)$ for which $-k>0$.
2. $\text{System is causal.}⟹h(n)=0(\forall n<0)$
   We proceed by taking the contrapositive of the statement above, such that $h(n)\ne 0(\forall n<0)⟹\text{System is not causal}$: