Consider the following system:
$$y(t-1)=\int_{-\infty}^\infty x()u(-t) d $$
where $u(t)$ is the unit step function, which is zero for $t<0$ and equals $1$ for $t>0$.
$(1)$ Is the system causal? Why or why not?
i think if $u(t)=0$ for all $t<0$. This means that $u(τ−t)=0$ for all $τ<t$ or, equivalently, for all $t>τ$ and the integrand is zero in range $({-\infty}, t)$.
Therefore, we can show that: $$\int_{-\infty}^\infty x()u(-t) d = \int_{t}^\infty x()u(-t) d = \int_{t}^\infty x() d$$ So, the system is not causal! am i right?!
$(2)$ Is the system time-invariant? Why or why not?
I'm a bit confused about $y(t-1)$ and lower bound ($t$) of integral, so I have nothing to say.