In this topic, you study the Causal & Non-Causal Systems theory, definition & solved examples.
Let $x(t)$ and $y(t)$ be the input and output signals, respectively, of a system shown in Figure 1. Then the transformation of $x(t)$ into $y(t)$ is represented by the mathematical notation
$y(t) = {\mathbf{T}}x(t)$
where $\mathbf{T}$ is the operator which defined rule by which $x(t)$ is transformed into $y(t)$.
Figure 1: System with a single input and output signal.
Causal System
A system is called causal if its output is independent of future values of input. Example of causal systems are
\[y(t) = x(t)\]
\[y(t) = x(t – 1)\]
\[y(t) = x(t) + x(t – 1)\]
Non-Causal System
A system is called noncausal if its output at the present time depends on future values of the input. Example of noncausal systems are
\[y(t) = x(t + 1)\]
\[y(t) = x(t) + x(t + 1)\]
Example : Determine whether or not each of the following systems are causal with input $x(t)$ and output $y(t)$.
(i) \[y(t) = x(3t)\]
(ii) \[y(t) = x(-t)\]
(iii) \[y(t) = {e^{x(t)}}\]
Solution : (i) \[y(t) = x(3t)\]
put $ t = 1$
\[ y(1) = x(3) \]
hence the system is Non-causal as output $y(1)$ depends on future input $x(3)$.
Solution : (ii) \[y(t) = x(-t)\]
put $t$ = – 1
\[ y(-1) = x(1) \]
hence the system is Non-causal as output $y(-1)$ depends on future input $x(1)$.
Solution : (iii) \[y(t) = {e^{x(t)}}\]
The system is causal as output $y(t)$ depends on present input $x(t)$ only.
