Question

Consider a continuous-time system with input-output relation :

$y\left(t\right)=\frac{1}{T}{\int }_{t-T/2}^{t+T/2}x\left(\tau \right)d\tau$

Select the correct option given below,

• A. Linear, time invariant, causal
• B. Linear, time varient, causal
• C. Linear, time invariant, non casual
• D. Non-linear, time invariant , causal

Answer 1

The correct option is C Linear, time invariant, non casual

Integration is linear system, Time-variant (or) time invariant system;
Delay input by $t_o$

$=\frac{1}{T}{\int }_{t-\tau /2}^{t+\tau /2}x(\tau -t_o)d\tau ....\left(i\right)$

Delay output by $t_o$ units (or) substitute $(t-t_0)$ in the place of t.

$y(t-t_0)=\frac{1}{T}{\int }_{t-l_0-\tau /2}^{t-t_0+\tau /2}x\left(\tau \right)d\tau ....\left(ii\right)$

From equation (i) and (ii), we can say equation (i) = equation (ii),
$\therefore$ The given system is time invariant,

Causal (or) Non-causal system:

$y\left(t\right)=\frac{1}{T}{\int }_{t-\tau /2}^{t+\tau /2}x\left(\tau \right)d\tau$

Let, T = 4

$y\left(0\right)=\frac{1}{4}{\int }_{-2}^{2}x\left(\tau \right)d\tau$

here, y(0) depends on future value x(2).

$\therefore$ The given system is non-causal system.