determine if the system is time-invariant

Question

I need to determine if the following system is time-invariant or not, and I'm a bit unsure about it. $$y(t)= \int_{-\infty}^{t-2} \tau \cdot x(2\tau)d\tau $$

Answer 1

The system is not time-invariant. Consider the response to the input signal $x_1(t)=x(t-t_0)$:

$$y_1(t)=\int_{-\infty}^{t-2}\tau x_1(2\tau)d\tau=\int_{-\infty}^{t-2}\tau x(2\tau-t_0)d\tau=\int_{-\infty}^{t-2-t_0/2}(\tau+t_0/2) x(2\tau)d\tau \neq y(t-t_0)$$

Answer 2

Let $x=1_{ [0,2] }$, then for $t > 100$, we have $y_{x}(t) = \int_{-\infty}^{98} \tau 1_{ [0,2] }(2 \tau) d \tau = \int_0^1 \tau d \tau = {1 \over 2}$.

If we let $x' = 1_{ [2,4] } $ (note that $x'(t) = x(t-2)$), then $y_{x'}(t) = \int_{-\infty}^{98} \tau 1_{ [2,4] }(2 \tau) d \tau = \int_1^2 \tau d \tau = {3 \over 2}$.

Hence the system is not time invariant.

If it was, we would have $y_{x'}(t) = y_x (t-2)$.