Linearity and stationarity of a system with integral

Question

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 , ( $t= t-t_0$ ).

Answer 1

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 $y(t-t_0)$.