The input x(t) and the output y(t) of a continuous-time system are related as
$y (t) = \int_{t - T}^{t} x (u) d u$
The system is

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Solution

$Given that, y (t) = \int_{t - T}^{t} x (u) d u$

Since the given system satisfies both homogeneity and additivity properties, the system is linear.

Check for time invariance:

$y (t - t_{o}) = \int_{t - t_{o} - T}^{t - t_{o}}$

When the applied input is $x (t - t_{o}),$

$y_{1} (t) = \int_{t - T}^{t} x (u - t_{o}) d u = \int_{t - t_{o} - T}^{t - t_{o}} x (τ) d τ$
$= y (t - t_{o})$

$\Rightarrow$ System is time invariant.

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Similar questions

\int_{- \infty}^{t} x (τ) c o s (3 τ) d τ

P_{1}

P_{2}

P_{3}

R_{1}

R_{2}

R_{3}

R_{4}

Properties:

P_{1}

P_{2}

P_{3}

Relations:

R_{1} : y (t) = t^{2} x (t)

R_{2} : y (t) = t | x (t) |

R_{3} : y (t) = | x (t) |

R_{4} : y (t) = x (t - 5)

e^{- | x (t) |}

y (t) = 5 t \int - \infty x (τ) d τ

y (t) = \int_{- \infty}^{t - 10} e^{- (t - 5 - τ)} x (τ) d (τ)

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