A system is causal if the output at time t depends on:

Only the present input, never past ones

In y(t) = x^2(t) + x(t+2), which term makes the system non-causal?

x(t+2), since it uses the input 2 seconds ahead

x^2(t), since squaring is non-linear

What does the non-causality of this system imply?

It cannot be built to work in real time

2081 Chaitra · 4 marks

Causality of y(t) = x²(t) + x(t + 2)

Short answer

The term x(t+2) requires knowledge of the input two seconds into the future at every t. Thus the output depends on future values, making the system non-causal (anticipative).

Determine whether the signal y(t) = x²(t) + x(t + 2) is causal or not.

A system is causal if the output at any time $t$ depends only on present and past values of the input, never on future values, i.e. $y(t_0)$ depends only on $x(t)$ for $t \le t_0$ .

Given: $y(t) = x^2(t) + x(t+2)$

Check the output at a few time instants:

$t$	$y(t)$	Input values needed
0	$x^2(0) + x(2)$	$x(0)$ present, $x(2)$ future
1	$x^2(1) + x(3)$	$x(3)$ future
$-2$	$x^2(-2) + x(0)$	$x(0)$ future

The term $x^2(t)$ uses the present input, which is allowed.
The term $x(t+2)$ uses the input 2 seconds ahead, i.e. a future value, at every instant $t$ .

Since the output at time $t$ needs $x(t+2)$ , which has not yet arrived, the system cannot be built to work in real time.

Answer: $y(t) = x^2(t) + x(t+2)$ is non-causal (anticipative). (Note also that, because of the $x^2(t)$ term, it is a non-linear system.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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