Causal and Non Causal Systems – Theory | Solved Examples

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 $y(t)$.

signal and system

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.

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