I just finished reading the fundamentals chapter about signals (linearity,causality,memory and time invariance). I wanted to solve some exercises and I found this one.
We have a signal with output described by $$y(t) = \int_t^{t+1}x(\tau-a)\;d\tau ,a\in R$$
- find if the system has memory, is stable, time invariant and linear
- which values of $a$ make the system causal
I am trying to understand things here. The system has memory if $a<0$ and memoryless $a>0$? I don't know from where to start to find the others.
EDIT
To prove linearity I did this
$$y(t)=F\{x(t)\}$$ $$F\{c_1x_1(t)+c_2x_2(t)\}$$ $$\int_t^{t+1}c_1x_1(\tau -a)+c_2x_2(\tau -a)d\tau $$ $$=\int_t^{t+1}c_1x_1(\tau -a)d\tau +\int_t^{t+1}c_2x_2(\tau -a)d\tau$$ $$=c_1y_1(t) + c_2y_2(t)$$
time invariance
$$F\{x(t-t_0)\}$$ $$=\int_t^{t+1}x(\tau -a-t_0)d\tau $$ $$=\int_{t-t_0}^{t+1-t_0}x(u -a)du $$ so $$=\int_{t-t_0}^{t+1-t_0}x(\tau -a)d\tau $$
which means it is time invariant
Am I right?