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How can I demonstrate the linearity and the stationarity of system with integrals? For example I have this signal $ y(t)= \int_{t}^{t-T} x(\tau ) d\tau $ and I know from theory that’s a filter so I should have both linearity and stationarity but I didn’t obtain this. For stationarity I first apply a delay on the input ( $x(t) = x(t-t_0)$ ) of the system and after on the output , ( $t= t-t_0$ ).

S.H.W
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Elena Martini
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1 Answers1

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Let $x_1(t)$ and $x_2(t)$ be inputs and $y_1(t)$ and $y_2(t)$ the corresponding outputs. We have $$z(t)= \int_{t}^{t-T} (ax_1(\tau )+bx_2(\tau )) d\tau = a\int_{t}^{t-T} x_1(\tau )d\tau + b\int_{t}^{t-T} x_2(\tau )d\tau = ay_1(t) + by_2(t)$$

So the system is linear. It's also time-invariant since if $x(t)$ is the input and $y(t)$ is the corresponding output then $$z(t)= \int_{t}^{t-T} x(\tau -t_0) d\tau = \int_{t-t_0}^{t-T - t_0} x(u) du = \int_{t-t_0}^{(t - t_0) - T} x(u) du = y(t-t_0)$$Which shows the output to the input $x(t-t_0)$ is $y(t-t_0)$.

S.H.W
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  • I don’t understand what is $ x(u) $ ? To find that is time-invariant first let’s say $ x(t-t_0) = x(\tau) $. So I obtained a y(t) where only x is changed. After I say that if $ y(t) = y(t-t_0) $ so I have to say that $ t=t-t_0 $ into the equation. But in this case I’ll have different extremes of integration. This is my doubt , but probably I made mistakes in the process ... – Elena Martini Sep 30 '20 at 07:17
  • @ElenaMartini According to the definition, the output to the input $x_1(t) = x(t-t_0)$ is $z(t)= \int_{t}^{t-T} x_1(\tau) d\tau$ which is equal to $z(t)= \int_{t}^{t-T} x(\tau - t_0) d\tau$ and by change of variable $u = \tau - t_0$ we have $z(t) = \int_{t-t_0}^{(t - t_0) - T} x(u) du$. This is obviously $y(t-t_0)$. – S.H.W Sep 30 '20 at 11:08
  • Ok , but about the extremes of integrations why we have ( after the change of variable ) $ \int_{ t-t_0}^{(t-t_0)-T } $. If $ u = \tau -t_0 $ we should only change the argument of integration and not the extremes t and t-T. This because $ u=\tau - t_0 $ and this is not related to $ t $ or $ T $ .. this is the part I don’t understand. Thank you so much – Elena Martini Sep 30 '20 at 12:06
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    @ElenaMartini You should change the limits of integration. See this for example. – S.H.W Sep 30 '20 at 12:43