Causal and Non Causal Systems – Theory | Solved Examples

In this topic, you study the Causal & Non-Causal Systems theory, definition & solved examples.

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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.