I have a system and I want to find which properties hold for this system
$$
y(t) = \left.\begin{cases} x(t)
& t \geq 1 \\ \\ 0 & -1 < t < 1 \\ \\
-x(t) & t \leq -1 \end{cases} \right\} \\ \\
$$
is the system linear, time invariant, causal , invertible, stable?
My problem is that I don't know how to work, because the function is piecewise defined.If I try to prove for example that each piece of function is linear then the whole function is linear ? With this methodology I found that the system is linear, causal and stable.Am I right?
Best Answer
The system is linear, causal, stable, but not time-invariant.
It is linear because from the definition if \$y_1(t)\$ is the response to input \$x_1(t)\$, and \$y_2(t)\$ is the response to input \$x_2(t)\$, then the response to the input signal \$ax_1(t)+bx_2(t)\$ is given by
$$y(t)=ay_1(t)+by_2(t)$$
It is causal because in order to compute the current output signal, not future values of the input signal are necessary, i.e. to compute \$y(t_0)\$ we only need to know \$x(t_0)\$ (and not even its past values, i.e. the system has no memory).
The system is stable in the BIBO (bounded-input bounded-output sense), because any bounded input signal \$|x(t)|<K\$ produces a bounded output signal \$|y(t)|<L\$ (in our case \$K=L\$).
The system is time-variant because the response to a shifted version of the input signal is not equal to the shifted output signal, i.e. if \$y(t)\$ is the response to \$x(t)\$, then \$y(t-t_0)\$ is generally not the response to \$x(t-t_0)\$. You can see this by noting that the output is always zero for \$-1<t<1\$, no matter how the input signal is shifted. If the system were time-invariant, then also the zero portion of the output would need to be shifted with the input signal.